design of an annular disc-shaped heat pipe for air-cooled
TRANSCRIPT
University of Central Florida University of Central Florida
STARS STARS
Electronic Theses and Dissertations, 2020-
2020
Design of an Annular Disc-shaped Heat Pipe for Air-cooled Steam Design of an Annular Disc-shaped Heat Pipe for Air-cooled Steam
Condensers Condensers
Ahmad Saleh University of Central Florida
Part of the Energy Systems Commons
Find similar works at: https://stars.library.ucf.edu/etd2020
University of Central Florida Libraries http://library.ucf.edu
This Doctoral Dissertation (Open Access) is brought to you for free and open access by STARS. It has been accepted
for inclusion in Electronic Theses and Dissertations, 2020- by an authorized administrator of STARS. For more
information, please contact [email protected].
STARS Citation STARS Citation Saleh, Ahmad, "Design of an Annular Disc-shaped Heat Pipe for Air-cooled Steam Condensers" (2020). Electronic Theses and Dissertations, 2020-. 619. https://stars.library.ucf.edu/etd2020/619
DESIGN OF AN ANNULAR DISC-SHAPED HEAT PIPE FOR AIR-COOLED STEAM
CONDENSERS
by
AHMAD SALEH
B.S. Mechanical Engineering, Jordan University of Science and Technology, 2008
M.S. Mechanical Engineering, Purdue University, 2012
A dissertation submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in the Department of Mechanical and Aerospace Engineering
in the College of Engineering and Computer Science
at the University of Central Florida
Orlando, Florida
Summer Term
2020
Major Professor: Jayanta S. Kapat
ii
© 2020 Ahmad Saleh
iii
ABSTRACT
Limitations on water utilization are turning into an expanding issue for the power
and electricity generation industry. As a contribution to the solution of water
consumption problems, utility companies are shifting toward using air-cooled
condensers (ACC) in replace to the typical water-cooling methods of once-through
cooling and the surface condenser/wet-cooling tower combination. Although the ACC is
a dry cooling method, the industry is quite hesitant to switch over to ACC mainly for
three reasons: (a) lower power output, (b) higher capital cost, and (c) larger physical
footprint. All these drawbacks are because of the high overall thermal resistance of
condensing steam to the ambient air compared to condensing it to water.
In this study, detailed mathematical equations were derived to model the heat
transfer process through the fined tubes of the ACC. The total thermal resistance model
was analyzed and investigated theoretically. The model was used to identify the design
components with the most significant effect on the overall thermal resistance of the ACC
system.
This study proposed a feasible cooling system based on heat pipe technology,
using a novel disc-shaped heat pipe design. The solution addresses the three problems
highlighted in using the air-cooled condensers in steam powerplant condensers. The
analysis covered design and manufacturing considerations, in addition to the thermal
iv
performance and the limitations of the proposed annular disc-shaped heat pipe. The
proposed annular disc-shaped heat pipe was investigated using three analysis
techniques. The first is a theoretical investigation of the heat transfer limitations of the
proposed annular disc-shaped heat pipe. This analysis was used to predict the capillary
and boiling thermal limitations of the proposed heat pipe design. Secondly, an annular
disc-shaped heat pipe was designed and built for the experimental investigation using
de-ionized water as the working fluid. The results obtained by the parametric analysis
were used as the input for the experimental design. Third, A detailed mathematical set
of equations was derived to model the heat pipe thermal resistance.
The experimental setup was validated by comparing the results to well-referenced
experimental results of similar disc-shaped heat pipe with different evaporator
configurations. The experimental results were compared to the thermal resistance model
developed in this study. The results showed a starting regime of the heat pipe, where the
thermal resistance is decreasing until it reaches a steady performance before it starts to
increase again when it reaches the heat transfer limits. The experimental results showed
a good agreement with the model prediction in the steady-state regime for heat inputs
over 300 w. The data identified two thermal performance regimes of the heat pipe, a
single-phase, and a two-phase regime. The second regime starts when the vapor region
reaches the isothermal state.
v
The heat pipe was tested with a maximum total heat input of 600 w, using three
types of wicks. The study presented the effect of the wick parameters such as porosity,
permeability, mesh number, and wire diameters on the performance. The impact of the
filling ratio was investigated by testing the heat pipe performance with 30 ml, 60 ml, 90
ml, and 120 ml filling amount.
The study showed high promising results for the proposed solution. It established
the theoretical and experimental validation of the system by validating the concept of the
annular disc-shaped heat pipe. Further investigation can be conducted starting from the
experimental results.
vi
DEDICATED TO MY FAMILY
vii
ACKNOWLEDGMENT
First and foremost, all praise and thanks are due to god for giving me the opportunity to
accomplish this work. I would like to express my sincere acknowledgment to my advisor, Dr.
Jayanta Kapat, for his mentorship, support, and guidance throughout my long journey at the
University of Central Florida. Words can not express my appreciation for all concern and support
during all the challenges and critical times in my professional and personal life. You were always
there with your guidance and support.
I would like to extend my appreciation for the members of my dissertation committee, Dr.
Jihua Gou, Dr. Subith Vasu, and Dr. Manoj Chopra, thank you for all the help and feedback
throughout this work.
I would like to thank all my colleagues at UCF and CATER for all their support during
these years. Thank you for making this journey more fun and enjoyable. I would like to extend my
acknowledgment to all my colleagues at Siemens Gamesa Renewable Energy for all the support
and understanding throughout this work.
I would like to extend my acknowledgment and appreciation to my Dad, Mom, brothers,
and sisters for all the support, encouragement, and love you always provided me over these years.
Finally, to the person that I would not even be able to do any of this work without, to my
soulmate and life partner, my lovely wife, Rana Amayreh, no words can express how much I am
lucky to have you in my life. Thanks to my precious kids, Laith and Talia, you are the joy of this
life.
viii
TABLE OF CONTENTS
LIST OF FIGURES ...................................................................................................................... xii
LIST OF TABLES ...................................................................................................................... xvii
LIST OF NOMENCLATURE ................................................................................................. xviii
CHAPTER 1: INTRODUCTION ................................................................................................ 1
Background ............................................................................................................................... 1
Air Cooled Systems (Dry Cooling) ........................................................................................ 3
Heat Pipes ............................................................................................................................... 10
CHAPTER 2: OBJECTIVES & MOTIVATION ....................................................................... 14
Objectives of the Present Study ........................................................................................... 14
Novelty and Intellectual Contribution ................................................................................ 15
CHAPTER 3: ANALYSIS OF AN ACC UNIT ....................................................................... 17
ACC Thermal Resistance Analysis ...................................................................................... 17
Heat Transfer Coefficients Correlations ............................................................................. 22
Condenser Steam-Side Temperature .................................................................................. 24
CHAPTER 4: PARAMETRIC ANALYSIS OF ANNULAR DISC-SHAPED HEAT PIPE 28
ix
Heat Pipe Thermal Resistance Analysis ............................................................................. 34
Heat Transfer Limitations ..................................................................................................... 44
Capillary Limitation .......................................................................................................... 45
Boiling Limit ....................................................................................................................... 54
CHAPTER 5: EXPERIMENTAL SETUP ................................................................................. 58
Annular Disc Heat Pipe Design ........................................................................................... 59
Heat Pipe Design Considerations .................................................................................... 59
Annular Disc-Shaped Heat Pipe Dimensions ................................................................ 60
Experimental Measurements ................................................................................................ 68
Temperatures ...................................................................................................................... 69
Pressure ............................................................................................................................... 72
Heaters ................................................................................................................................. 75
Experimental Heat Loss .................................................................................................... 76
Vacuum and Charging Station ............................................................................................. 79
Test Section ............................................................................................................................. 81
Test Procedure ........................................................................................................................ 82
x
Data Reduction ....................................................................................................................... 83
Uncertainty Analysis ............................................................................................................. 85
CHAPTER 6: RESULTS ............................................................................................................. 88
Experimental Results ............................................................................................................. 88
Repeatedly Test .................................................................................................................. 89
Model and Experimental Validation ............................................................................... 92
Vapor Temperature ......................................................................................................... 105
Evaporator Temperature ................................................................................................. 110
Condenser Temperature ................................................................................................. 112
Thermal Resistance .......................................................................................................... 118
CHAPTER 7: CONCLUSION AND FUTURE WORK ....................................................... 121
APPENDIX A: NUMERICAL ANALYSIS OF CONVENTIONAL HEAT PIPE ............ 124
NUMERICAL SETUP .......................................................................................................... 125
Numerical Model ................................................................................................................. 128
Vapor Region .................................................................................................................... 129
Porous-Wick Region ........................................................................................................ 130
xi
Solid Container ................................................................................................................. 131
CFD Model Validation ........................................................................................................ 131
APPENDIX B: WIND TUNNEL AIR-FLOW MEASUREMENTS .................................... 133
LIST OF REFERENCES ........................................................................................................... 136
xii
LIST OF FIGURES
Figure 1 T-S diagram of the ideal Rankine cycle [16] ............................................................. 6
Figure 2 Characteristic curve of a steam turbine [18] ............................................................. 7
Figure 3: Schematic of (a) full ACC condenser, (b) one ACC cell, (c) single finned tube.
[19] .................................................................................................................................................. 8
Figure 4: ACC street with 6 Cells [20]. ...................................................................................... 9
Figure 5: Conventional heat pipe schematic [22] .................................................................. 11
Figure 6: Operating temperature range of conventional working fluids [28] ................... 13
Figure 7: Geometry of plate-fin heat exchanger with flattened channels [30]. ................. 17
Figure 8: Heat Transfer path in a finned tube ........................................................................ 19
Figure 9: Equivalent thermal resistance circuit of a finned tube ........................................ 19
Figure 10: Typical heat pipe performance map [22] ............................................................. 29
Figure 11: Proposed Air-cooled condenser design using annular disc-shaped heat pipes
....................................................................................................................................................... 31
Figure 12: A cross-sectional view of an annular disc-shaped heat pipe [31] .................... 33
Figure 13: Annular disc-shaped heat pipe thermal resistance map. .................................. 36
Figure 14: CFD velocity distribution over the annular-disc shaped heat pipe ................. 41
Figure 15: CFD - Heat transfer coefficient over the condenser face ................................... 42
xiii
Figure 16: Calculated average heat transfer coefficient over the condenser external surface
....................................................................................................................................................... 42
Figure 17: Effective thermal resistance of the ADHP ........................................................... 44
Figure 18: Pressure variation in a heat pipe ........................................................................... 47
Figure 19: Variation of Capillary maximum pressure, thermal conductivity, porosity, and
permeability vs. the wick mesh number. ............................................................................... 50
Figure 20: variation of permeability, thermal conductivity, and maximum capillary
pressure with the porosity of metal screen mesh. ................................................................. 52
Figure 21: effect of vapor temperature and nucleation radius on the boiling limit ......... 57
Figure 22: Cutaway view of the annular disc-shaped pipe ................................................. 61
Figure 23: Condenser copper plate dimensions .................................................................... 62
Figure 24: Sidewall of the annular disc-shaped heat pipe ................................................... 62
Figure 25: Dimensions of the center copper cylinder (evaporator) .................................... 63
Figure 26: Mechanical lock assembly of the heat pipe components ................................... 64
Figure 27: Annular disc-shaped heat pipe with an open-top .............................................. 65
Figure 28: Built heat pipe CAD ................................................................................................ 67
Figure 29: Cross-section A-A from Figure 28 ......................................................................... 67
Figure 30: Location of the evaporator thermocouples .......................................................... 70
Figure 31: locations of the condenser surface thermocouples ............................................. 71
xiv
Figure 32: location of vapor temperature and pressure measurements ............................ 73
Figure 33: Pressure transducer calibration curve vs. MKS transducer .............................. 74
Figure 34: location of temperature measurement in the insulation cylinders. ................. 78
Figure 35: Variation of heat loss percentage with total heat input ..................................... 78
Figure 36: schematic of the vacuuming and charging station ............................................. 80
Figure 37: CAD of the wind tunnel and test section ............................................................. 81
Figure 38: Transient average evaporator temperature for 120 ml water filling with 5 layers
of 145 copper screen mesh (Case 2C) ...................................................................................... 89
Figure 39: Transient absolute vapor pressure for 120 ml water filling with 5 layers of 145
copper screen mesh at 600 w (Case 2C) .................................................................................. 90
Figure 40: Repeatedly test - variation of vaportemperature with heat input for 90ml water
with with 5 layers of 145 copper screen mesh (Case 2B). ..................................................... 91
Figure 41: Repeatedly test - variation of average evaporator temperature with heat input
for 90ml water with with 5 layers of 145 copper screen mesh (Case 2B). .......................... 92
Figure 42: Experimental validation of the current study with North [80]. ........................ 94
Figure 43: Performance curve of the empty heat pipe for the current study and North [1]
....................................................................................................................................................... 96
Figure 44: schematic of the fin analysis - top condenser cross-section .............................. 97
xv
Figure 45: Condenser temperature distribution at the parallel and perpendicular radial
axes – analytical vs. experimental (Case 0). ........................................................................... 99
Figure 46: Radial condenser temperature distribution of empty heat pipe – analytical vs.
average experimental (Case 0). .............................................................................................. 100
Figure 47: Heat pipe effective and total thermal resistance for 120 ml charged heat pipe
with N=200 copper wick ......................................................................................................... 102
Figure 48: Heat pipe effective and total thermal resistance for 90 ml charged heat pipe
with stainless steel wick .......................................................................................................... 102
Figure 49: Thermal resistance model validation vs current experimental and North [81].
..................................................................................................................................................... 104
Figure 50: schematic of the fin analysis for North’s design - bottom condenser plate .. 105
Figure 51: Temperature locations in the vapor region ....................................................... 105
Figure 52: Transient vapor pressure variation over the radial axis .................................. 106
Figure 53: One-minute average vapor temperature over the radial axis ......................... 107
Figure 54: Top view of the heat pipe with Acrylic top plate Case 1B .............................. 108
Figure 55: effect of non-condensable gases on the vapor region (run 1 @1.5 KPa, run 2 @6.3
KPa initial pressure) ................................................................................................................ 109
Figure 56: evaporator axial temperature distribution for case 4C .................................... 110
Figure 57: evaporator circumferential temperature distribution for case 4C ................. 112
xvi
Figure 58: Radial temperature distribution of the bottom plate at θ = 0, (Case 2C) ...... 114
Figure 59: Radial temperature distribution of the bottom plate at θ =π/2 (Case 2C) .... 114
Figure 60: Top plate radial temperature distribution at θ = 0 (Case 2C) ......................... 115
Figure 61: Top plate radial temperature distribution at θ = π/2 (Case 2C) ..................... 115
Figure 62: Surface average temperature of the top and bottom condenser plates vs. the
total heat input (Case 2C) ....................................................................................................... 116
Figure 63: variation of heat pipe effective thermal resistance with wick type ............... 118
Figure 64: ADHP effective thermal resistance at different filling ratios. ......................... 119
Figure 65:ADHP thermal resistance variation with heat input. (case 1C and 1B) ......... 120
Figure 66: Schematic of the heat pipe and the coordinate system .................................... 126
Figure 67: CFD temperature distribution validation .......................................................... 132
Figure 68: Wind-Tunnel air-flow measurement grid ......................................................... 134
Figure 69: Wind-tunnel local traversed mass flow ............................................................. 135
Figure 70: Wind-tunnel local traversed mass flow ............................................................. 135
xvii
LIST OF TABLES
Table 1: Classifications of heat pipes by operating temperature [28] ................................. 13
Table 2: Comparative values for heat pipe thermal resistances .......................................... 37
Table 3: Expressions for wick effective capillary radius and permeability ....................... 48
Table 4:Variation of screen wick properties with its mesh number ................................... 49
Table 5: Variation of screen wick properties with porosity ................................................. 52
Table 6: Parameter values for the boiling limit. ..................................................................... 57
Table 7: wick specifications ...................................................................................................... 66
Table 8: Material of the built ADHP ........................................................................................ 68
Table 9: Pressure transducer calibration measured values .................................................. 74
Table 10: Insertion heaters specifications ............................................................................... 76
Table 11: Experimental uncertainty values ............................................................................ 87
Table 12: configuration of the experimental cases. ............................................................... 88
Table 13: Standard error of the condenser mean temperature .......................................... 117
Table 14: Heat pipe operation’s flow chart and its interaction between different regions.
..................................................................................................................................................... 127
Table 15: Wind-Tunnel measurements ................................................................................. 134
xviii
LIST OF NOMENCLATURE
A = Cross-sectional area (m2)
Cp = Specific heat (KJ/Kg.K
D = Diameter (m)
Dh = Hydraulic diameter (m)
hfg = Latent heat (J/kg)
h = Heat Transfer Coefficient (W/m2-K)
K = Permeability (m2)
k = Thermal conductivity (W/m-K)
= Mass flow rate (kg/s)
N = Mesh number (in-1]
Nu = Nusselt number
po = Total pressure (Pa)
p = Pressure (Pa)
Pr = Prandtl Number
R = Resistance (Ω)
R = Thermal resistance (k/w]
Re = Reynolds number
Rair = Gas constant of air
xix
r = Radius (m), radial coordinate.
T = Temperature (°C)
t = Thickness (m)
U = Velocity (m/s)
SD = Standard deviation
SEM = Standard error from mean
Subscripts
c, cond = Condenser
e, evap = Evaporator
eff = Effective
f = Fin
l = liquid
s = Solid
v = vapor
w = wick
Greek Symbols
η = Thermal efficiency
xx
휀 = Thermal effectiveness
휀 = Porosity
𝜎 = Fluid surface tension
ρ = Density (kg/m3)
= Dynamic Viscosity
= Kinematic Viscosity
θ = Tangential coordinate
Acronyms
AACC = A-frame air cooled condenser
AC = Air cooled
ADHP = Annular disc-shaped heat pipe.
CFD = Computational Fluid Dynamics
1
CHAPTER 1: INTRODUCTION
Background
Limitations on water utilization are turning into an expanding issue for the power
and electricity generation industry. The water withdrawal and consumption in the power
generation industry and its environmental effect has been widely studied in the literature
[1-11]. The U.S Geological Survey (USGS) [2] has reported that 41% (540 Mm3/day) of the
U.S. total freshwater withdrawals are used in electricity production. Most of the water in
thermoelectric production is used in once-through coolers, where the water is returned
to its source. The energy production accounts for about 3.3% (12.5 Mm3/day) of total
water consumption in the United States [9].
All the power plants that are based on the Rankine cycle, either in simple or
combined cycles, require heat rejection to the environment to condense the turbine
exhaust steam[12]. The heat rejection in modern combined-cycle ranges from 40% of the
fuel energy and up to 70% for a traditional Rankine-cycle [13]. Typically, there are three
major condensation systems used for this heat rejection:
• Once-through water cooling-steam surface condensers.
• Combined systems of a wet-cooling tower and steam surface condensers.
• Air-cooled condensers systems (Dry-cooled condensers).
2
The once-through cooling systems are still utilized in over 30% of today’s fleet of
power plants. In once-through cooling systems, the water from an environmental source
such as near-by lake or river is cycled through the condenser of the turbine exhaust steam.
The water is then returned to the same source warmer by about 20°F [1]. In between the
commercially used cooling methods, this method is considered the most inexpensive.
However, the resulting increase in temperature of the cycled water creates an
environmental concern. Another disadvantage of using this method is the cost of
maintenance to clean the tubes of fouling to maintain its performance.
In the combined surface condenser with a cooling tower system, the turbine
exhaust steam condenses on the outside surface of a bundle of tubes by extracting the
heat to the cold water circulated inside the tubes. After receiving the rejected heat from
the condensed steam, the circulated water is piped to the top of a cooling tower. And it
flows downward through fill material that breaks the water up into droplets or spread it
out into a thin film to maximize the surface area exposed to the cooling air, which is
drafted through the tower by natural convection or large fans. The cooling method in
these towers is considered wet cooling, where its cooling capacity is limited by the
cooling air wet-bulb temperature (TWB). This type of cooling system can maintain
consistent exhaust pressure for the turbine. However, this type of cooling system
consumes a large amount of water. About 80 to 90% of the rejected heat is released to the
3
atmosphere through the evaporation of water. Thus, this system requires a large amount
of make-up water to keep the condenser working on the required performance [1]. In this
type of cooling system, about 0.47 gallons (1.8 L) of water are consumed per kilowatt-
hour of electricity consumed at the point of end-user[10].
Although the once-through cooling systems withdraw up to 100 times more water
per unit of electric production than the cooling towers technologies, the cooling tower's
systems consume at least twice as much water as once-through cooling systems [7].
Air Cooled Systems (Dry Cooling)
Every year more restrictions on utilization of surface or groundwater are added
and imposed by communities around the country, which affect the cost of electricity
production. As a contribution to the solution of water consumption problems, utility
companies are shifting toward using air-cooled condensers (ACC) in replace to the
typical water-cooling methods of once-through cooling and the surface condenser/wet-
cooling tower combination. Although the ACC is a dry cooling method, the industry is
quite hesitant to switch over to it, that’s mainly due to three reasons:
(a) lower power output
(b) higher initial cost.
(c) larger physical footprint.
4
All these drawbacks of the air-cooled condenser are because of the high overall
thermal resistance of condensing steam to the ambient air compared to condensing it to
water, due to the poor thermal conductivity of air[14].
In the dry cooling systems, the heat extracted from the exhausted steam is
ultimately rejected to the ambient air-driven across finned-tube heat exchangers. This
process typically is done by using one of two main types of dry systems: direct or indirect
dry cooling. In the indirect dry cooling systems, the steam condenses in a surface
condenser, like the once-through and recirculating systems process. However, the heated
cooling water is then cooled down by circulating it in an air-cooled heat exchanger. In the
direct system, the steam is condensed in the air-cooled condenser by directing the exhaust
steam directly into the ACC. In these two dry systems, the air is driven by mechanical or
natural draft units. All the steam dry cooling systems in the United States are using the
direct system with a mechanical-draft ACC [15].
In contrast to the steam surface condensers, the exhaust steam in the air-cooled
condenser (ACC), condenses inside an of arrayed cells of finned tubes by extracting the
heat to the continual external airflow outside the tubes. Thus, the primary heat transfer
mode in the process is by convection without any water consumption as there is no
evaporation process; that is why it is referred to as Dry cooling. Because of that, the
condensing steam temperature is theoretically limited by the external air dry-bulb
5
temperature (TDB). As the dry-bulb temperature always equals or higher than the wet-
bulb temperature, the dry-cooling methods generally provide higher exhaust pressure
for the turbine compared to the water-based systems. The higher steam condensing
temperature penalizes the cycle efficiency, particularly on hot days with low humidity as
the condenser increases the steam temperature to be able to extract the required amount
of heat to hot ambient temperature.
Figure 1 depicts the T-S diagram of an ideal Rankine cycle, typically the heat
transferred to the water in the boiler is represented by the area under process curve 2-3,
and the heat rejected in the condenser is represented by the area under the process curve
4-1. The difference between these two areas sums up the total net-work produced in the
cycle. The cycle work increases, which is the integral of T.ds around the cycle, by
decreasing the exit pressure from P4 to P4’. As a result of that, the mean temperatures at
which the heat is absorbed and rejected both decreases, the most substantial change is in
the mean temperature of the heat rejection, which increases the cycle thermal efficiency.
6
Figure 1 T-S diagram of the ideal Rankine cycle [16]
Figure 2 shows a typical characteristic curve of a steam turbine. From the curve,
the power generated from the turbine decreases with the increase in turbine exhaust
temperature. However, the higher the exhausted temperature, the higher the heat
rejection required. That is due to the reduction in the turbine efficiency with increasing
the turbine back pressure and condensing temperature. In general, the condenser must
be designed according to the turbine power and the ambient conditions to have enough
cooling capacity under all operating conditions. When ambient conditions reduce the
efficiency of the condenser, the steam temperature must increase to reject the needed
heat; and less power is generated [7].
7
For a 500 MW steam power plant, reducing the steam condensing temperature by
15°C from 50 °C to 35 °C would give an estimation of 5% more power production, that’s
equal to about $11M more annual income [17].
Figure 2 Characteristic curve of a steam turbine [18]
The air-cooled condenser (ACC) systems consist of many air-cooled heat
exchangers arranged in an A-frame configuration, as illustrated in Figure 3. The turbine
exhaust steam is ducted through a distribution manifold located above a row of finned
tubes. A typical full-scale ACC consists of several such rows referred to as “streets” or
“lanes.” Each row consists of three to six primary condenser cells connected in series with
secondary reflux (dephlegmator) condenser, as shown in Figure 4.
8
Figure 3: Schematic of (a) full ACC condenser, (b) one ACC cell, (c) single finned tube. [19]
In the A-frame ACC system, the finned tubes are typically about 30–40 ft long and
grouped in 8 ft bundles across. Typically, one cell has plan dimensions of 40 ft x 40 ft,
which contains around five bundles on each face, with a total of 10 bundles per cell. Many
different geometries of finned-tube geometries have been used over the years, such as
circular tubes with wrapped, round fins, and elliptical tubes with plate fins. Finned tubes
are usually arranged in two to four rows in a staggered array. However, the most recent
ACC designs have used elongated flow tubes separated by plate fins, referred to as a
single row condenser (SRC) [15].
As an example, a 500-MW combined-cycle plant, typically have 30–40 cells to
condense an approximate of 125–150 kg/s of steam. Cells are usually arranged in a (5 x
6), (8 x 5), or two (4 x 5) layouts. That makes the total footprint of the ACC about 200 ft x
9
250 ft. adding to that bracing and vertical steel columns to support the cells and fans. The
fan deck is usually 60–80 ft above ground level, makes the large steam manifold at the
top of the cells at about 100–120 ft above grade.
The NSF/EPRI Joint Solicitation Informational Webcast conducted, 2013, reported
the average coast of a cell of 12x12 m2 footprint size, to be about $1.5 Million/ACC cell.
Figure 4: ACC street with 6 Cells [20].
10
Heat Pipes
A heat pipe is a closed two-phase heat transfer device that can passively transfer
a large amount of heat energy over a small temperature difference. The heat pipe in its
conventional design was introduced by Gaugler in 1944, when he introduced the use of
the wick structure, giving the heat pipe the ability to work against gravity. However, the
full potential of the heat pipe design was not realized by the scientific community until it
was highlighted by Grover research in 1964, where he showed that the heat pipe has a
higher thermal conductivity than any known metal. [21].
The heat pipe is a sealed device that contains both the liquid and vapor phases of
a working fluid. The device is heated from one end, called the evaporator, and cooled
from the other end, known as the condenser side. heat is applied at the so-called
evaporator section, and the phase change occurs, the vapor flows through the heat pipe
due to the pressure difference generated by the evaporation process, the heat is then
extracted at the other end, where the vapor condenses back to the liquid phase, the so-
called condenser section, The liquid then flows back to the evaporator section under the
influence of gravity or by a capillary structure passively completing the heat transfer
cycle
11
Figure 5: Conventional heat pipe schematic [22]
Figure 5 shows a schematic of a conventional heat pipe showing its operational
concept of heat and working fluid circulation. The heat pipe has been used in many
different configurations, with single or multiple heat sources or sinks with or without
adiabatic sections. A significant advantage feature of the heat pipe is its flexibility to be
designed according to the specific application [23].
Heat pipes have been used in many variant applications. However, it is well used
in space applications heating or cooling in vehicles, electronics cooling, and fuel cells. The
heat pipe low maintenance cost and its ability to efficiently transmit heat over low driving
temperatures make it an attractive choice for Multi-cascade machines [23-26]. And its
ability to transfer high heat flux rates makes it preferable for electronic cooling where the
footprint size is challenging [14, 27]
12
Heat pipes can be classified into different categories based on their geometries,
applications. The two primary ways to classify the heat pipes are based on either its
working fluid operating temperature, or the control mechanism of driving the liquid
between the condenser and evaporator (wick, gravitational, centrifugal, electrostatic, or
osmotic) [23]. Different applications have different operating conditions and different
temperature range [28]. Therefore, choosing a suitable working fluid is crucial. The
working fluid selection must consider the operating temperature and pressure, and the
compatibility between the working fluid and the heat pipe container and wick materials.
The working temperature range for the commonly used working fluids in heat pipe
applications are presented in Figure 6. heat pipes can be classified into four different
types based on its operating temperature, Table 1 summarize this classification. [28]
For the proposed application in the steam condenser for steam power plants,
several working fluid options are suitable for the working temperature range, such as
methanol, ammonia, acetone, and water. For its availability, well-established properties,
and compatibility with most of the casing materials, the de-ionized water was selected
for this study as the working fluid.
13
Table 1: Classifications of heat pipes by operating temperature [28]
type Temp. Range Specifications
High Temperature >700 K Using liquid metals, Potassium, Sodium, and silver
medium Temperature 550-700 K organic fluids such as Naphthalene and Biphenyl
Room Temperature 200-550 K typically, methanol, ethanol, ammonia, acetone, and
water
Cryogenic (Low
Temperature)
1-200 K fluids such as helium, argon, neon, nitrogen, and oxygen
Figure 6: Operating temperature range of conventional working fluids [28]
14
CHAPTER 2: OBJECTIVES & MOTIVATION
Objectives of the Present Study
Reflecting upon the review of literature from Chapter 1, water consumption in
power generation is a high raised concerned in the industry. The commercially available
type of air-cooled condensers for steam turbine applications addresses this problem and
provides a low water consumption solution. However, this solution penalizes the
performance of the steam turbine cycle due to its relatively low heat transfer efficiency.
This study aims to introduce a novel solution to this problem using a novel air-cooled
condenser, employing the heat pipe technology to increase the thermal of the steam
condensers. In general, the objective of this study is to identify the current challenges that
constrain the thermal performance of the air-cooled condenser and to introduce the
concept of the proposed heat pipe condenser while addressing all the design and
performance challenges that affect the proposed solution. The deliverables and measures
of study in the effort to achieve these objectives are outlined below:
1. Thermal analysis of the state-of-art air-cooled condenser
a. Thermal resistance analysis of the fin-tube structure.
b. Correlations of the heat transfer coefficients of the air and steam
sides.
15
c. Predicting the condenser’s pressure drop and heat transfer
performance.
d. Defining the main effective parameters on the condenser
performance.
2. Conceptual design of the annular disc-shaped heat pipe.
3. An analysis study of an annular disk-shaped heat pipe.
a. Thermal resistance analysis and 1D modeling.
b. Define the heat transfer limitations of the design.
c. Numerical modeling of the heat pipe process.
4. Experimental design and validation of the derived model.
5. Compare the proposed heat exchanger performance to the current state-of-art
steam turbine air-cooled condenser.
a. Heat transfer performance.
b. Reduction of the turbine backpressure of the cycle.
Novelty and Intellectual Contribution
Following are aspects of the various ways the current study is unique and novel:
- Air-cooled condenser analysis considering both air and steam sides.
- Novel Annular Disc-Shaped Heat Pipe design (ADHP).
- Thermal resistance analysis of ADHP.
16
- Modeling heat transfer limitations of ADHP.
- Experimental study of the ADHP under steady and transient states.
The proposed Annular disc-shaped heat pipe has a novel design that has not been
discussed in the literature before. The working concept of it can be thought of as a
combination between the flat or disc-shaped heat pipe and the concentric heat pipes
designs. The evaporator has the same concept of the typical annular heat pipe with the
heat transmit radially. The disc-shaped heat pipe has been introduced and analyzed by
Vafai [29]. However, this study proposes a new design for the evaporator side. The new
design makes it suitable for the proposed application of steam condenser.
This study provides a detailed analysis of a general model to predict the annular
disc-shaped heat pipe thermal performance with the consideration of the design shape,
heat pipe material, working fluid types, and condenser cooling method.
The study provides an extensive experimental investigation for the novel
proposed heat pipe that provides a base for further investigation and design optimization
in the effort to achieve the comprehensive solution to one of the well stated technical
challenges in power generation.
17
CHAPTER 3: ANALYSIS OF AN ACC UNIT
ACC Thermal Resistance Analysis
This chapter presents the analysis of the performance of a typical air-cooled
condenser unit (ACC). The analysis considers the most recent type of finned tubes used
in ACC, starting with the determination of its heat transfer features. Figure 7 depicts the
main geometrical parameters of a flattened finned-tube, which is the most recent design
used in ACC [30]. The analysis described in this study was presented by the author in
previous work [31].
Figure 7: Geometry of plate-fin heat exchanger with flattened channels [30].
18
In the Rankin cycle, the turbine backpressure depends on the condenser
temperature, which is controlled by the overall conductance between the steam side and
the cooling air passing over the fined tubes. This analysis provides a comprehensive
methodology for predicting the overall heat transfer coefficient for the fin-and-tube heat
exchanger. The outside heat transfer coefficient is obtained by a series of correlations for
laminar/turbulent single-phase flow in between the plate fins. On the other side, the two-
phase condensation region is also predicted by correlations for steam condensation inside
the tubes. The two sides, external single-phase and internal single-phase or two-phase
flows, were combined to predict the condenser temperature and heat transfer
performance.
A sketch of a cross-sectional view of a finned-tube is illustrated in Figure 8; the red
arrows indicate the heat transfer path between the steam inside the tube and the external
cooling air. Figure 9 illustrates the thermal resistance network for the cross-section under
investigation.
19
Figure 8: Heat Transfer path in a finned tube
Figure 9: Equivalent thermal resistance circuit of a finned tube
20
the fin tube total thermal resistance is obtained from:
𝑅𝑡𝑜𝑡𝑎𝑙 =1
𝑈𝑖𝐴𝑖=
(𝑇𝑠 − 𝑇𝑎)
𝑄 (1)
1
𝑈𝑖𝐴𝑖=
1
ℎ𝐶𝐴𝑖𝑛𝑡+ 𝑅𝑤 + (ℎ𝐵𝐴𝐵 +
1
𝑅"𝐴𝑐𝑜𝑛𝑡
+1
𝜂𝐹ℎ𝐹𝐴𝐹
)
−1
(2)
1
𝑈𝑖𝐴𝑖=
1
ℎ𝐶𝐴𝑐+
𝑡𝑤
𝑘𝐴𝑐+
1
𝜂0ℎ𝑎𝐴𝑡𝑜𝑡 (3)
Where R” is the contact resistance, Rw is the tube wall resistance, and η0 is the overall
surface efficiency calculated by:
𝜂0 = 1 −𝐴𝑓
𝐴𝑡𝑜𝑡(1 −
𝜂𝑓
𝐶1) (4)
𝐶1 = 1 + 𝜂𝑓ℎ𝑎𝐴𝑓 (𝑅"
𝐴𝑐𝑜𝑛𝑡) (5)
Aint = tube surface area, AB = un-finned surface area
Af = tube area, Acont = fin-tube contact area
Atot = air-side total area, Rw=tube wall resistance
R” = contact thermal resistance, ha = air side heat transfer coefficient
ηf =fin efficiency, η0 = air-side overall efficiency
hc = steam side heat transfer coefficient.
Typically, the conductance resistance in the fin is much smaller than the conviction
resistance. Hence, equation 3 can be simplified as:
21
𝑡𝑤
𝑘𝑝𝐴𝑐≪
1
ℎ𝐶𝐴𝑐, 𝑎𝑛𝑑 ℎ𝐵
𝐴𝐵
𝐴𝑓≪
1
𝑅"𝐴𝑓
𝐴𝑐𝑜𝑛𝑡+
1𝜂𝐹ℎ𝐹
(6)
The overall heat transfer coefficient can be written as:
1
𝑈𝑖=
1
ℎ𝐶+ 𝑅"
𝑆
𝑡𝑓+
𝑆
𝜂𝐹ℎ𝐹(2𝐿𝑓 + 𝑡𝑓) (7)
The analytical solution for a rectangular fin efficiency is [32]:
𝜂𝐹 =tanh(𝑚𝐿𝑒)
𝑚𝐿𝑒 (8)
Where,
𝑚 = √2ℎ𝑓
𝑘𝑓𝑡𝑓, 𝑎𝑛𝑑 𝐿𝑐 = 𝐿𝑓 +
𝑡𝑓
2 (9)
From the general derived analysis of the finned tube heat exchanger, the total
thermal resistance strongly depends on the condensing heat transfer coefficient, the fin-
tube contact resistance, and the fin thermal efficiency. The recent manufacturing
technology dramatically reduced contact resistance. However, the use of these
technologies is still costly to be implemented commercially for mass production and
requires a lot of effort and accuracy [33-35].
Some methods investigated in the literature, such as internal tabulators and
internal fins, showed an enhancement of the condensing heat transfer rate [36-40].
However, it is commercially challenging to use such methods due to the small internal
22
diameters and the very long tubes, which make the process economically not efficient.
The fin thermal resistance has been intensively studied in the literature to optimize the
geometry, size, and spacing [41-44]. Which makes the current implemented fin-tube
configuration, is the optimal design considering performance, coast, and manufacturing.
Further studies suggested using guided solid and slotted plates to enhance the external
heat transfer coefficient between the fins [45-48]
The proposed solution in this study considers the three components of the thermal
resistance and provide an improvement of the overall thermal performance of the
condenser by implementing the heat pipes technology in the power plant condensers.
Heat Transfer Coefficients Correlations
The heat transfer coefficient in the external airflow between the fins is modeled as
concurrently developing flow throw a channel. Following the same analysis presented in
our previous work [31], using the Nusselt number expression proposed by Sparrow [49]
and presented by Teertstra et al. [50], the developing flow region is expressed as:
𝑁𝑢𝑑𝑒𝑣 = (0.664
√𝐿𝑡ℎ∗ 𝑃𝑟1/6
) (1 + 7.3√𝑃𝑟𝐿𝑡ℎ∗ )
1/2 (10)
The developing region dimensionless thermal entrance length (𝐿𝑡ℎ∗ ) is given by:
23
𝐿𝑡ℎ∗ =
𝐿
𝐷ℎ𝑃𝑟𝑅𝑒𝐷𝐻 (11)
The heat transfer coefficient in the fully developed flow region between the flat
passages depends on the fin’s geometry and design. The Nusselt number is obtained
using the expression developed by Shah and Sekulic [51]:
𝑁𝑢𝑓𝑑 = 7.541(1 − 2.61 (𝑏
𝐻) + 4.97 (
𝑏
𝐻)
2
− 5.119 (𝑏
𝐻)
3
+2.702 (𝑏
𝐻)
4
− 0.548 (𝑏
𝐻)
5
(12)
Equations (10) and (12) represent the developing and fully developed regions,
respectively. The two expressions are combined using the addition of asymptotes method
to obtain the overall Nusslet number of the flow over the fins [52].
𝑁𝑢𝑎 = [(𝑁𝑢𝑑𝑒𝑣)𝑛 + (𝑁𝑢𝑓𝑑)𝑛
]1/𝑛
(13)
n is the blending parameter that controls the behavior of the model in the transition
region. The value for the blending parameter used in this model (n=3), as determined by
Teertstra et al. [50]. Thus, the heat transfer coefficient can be evaluated from
ℎ𝑎 =𝑁𝑢𝑎𝑘𝑓
𝐷ℎ (14)
The accuracy and applicability of Equation (14) were experimentally validated in [53].
Kroger [54] derived an analytical solution for the condensing heat transfer
coefficient in inclined finned flattened tubes
24
ℎ𝑐 = 0.9245 [𝑘3𝜌2𝐿𝑡𝑔 cos 𝜃 ℎ𝑓𝑔
𝜇𝑚𝑎1𝐶𝑝𝑎(𝑇𝑣 − 𝑇𝑎𝑖) [1 − exp −𝑈𝐴𝑐
𝑚𝑎𝐶𝑝𝑎]]
0.333
(15)
Where ma1 is the mass flow rate of air flowing on one side of the finned tube, hfg is
the latent heat of the steam. The overall heat transfer coefficient is approximated by the
effective airside heat transfer coefficient based on the condensation surface area of the
flattened tube, ignoring the film resistance.
Kroger’s analysis applies to that part of the tube where vapor velocity and the
shear stress on the condensate film are negligible. The shear stress on the condensate film
strongly affects the development of the film region, especially at the inlet region to the
relatively high velocity. However, gravity control becomes more critical further from the
inlet, and the above correlation gives a good approximate condensation heat transfer
coefficient for long inclined tubes.
Condenser Steam-Side Temperature
The steam condensing side of the air-cooled condenser has a high effect on
predicting the condenser performance. However, most of the current models in the
literature seem to be based solely on airside analysis, disregarding the steam-side
resistance for quantifying condenser performance could lead to erroneous results, in
particular when presented in terms of plant output.
25
In this study, a detailed ACC model is presented to evaluate the condenser
temperature and pressure. The model considers the airside thermal resistance and the
steam-side one.
In the derived model, isothermal heat rejection is assumed inside the steam
condenser, and neglecting the sensible heat rejection. Additionally, the model does not
consider any undesirable air-leakages or tube fouling in the condenser.
Upon these assumptions, the total heat transfer inside the condenser is given by:
= 𝑠ℎ𝑓𝑔 (16)
All the energy rejected from the steam must be transferred to the air crossing the heat
exchanger, to satisfy the energy balance on the air-cooled condenser under the stated
assumptions,
= 𝑎𝐶𝑝𝑎(𝑇𝑎𝑜 − 𝑇𝑎𝑖) (17)
Where ms and ma are the condensed steam and air mass flow rates respectively, hfg is the
steam’s latent heat, Tao and Tai is the outlet and inlet temperatures of the air. From the
equations (16) and (17) the air temperature at the exchanger exit can be expressed as:
𝑇𝑎𝑜 =𝑠ℎ𝑓𝑔 + 𝑎𝐶𝑝𝑎𝑇𝑎𝑖
𝑎𝐶𝑝𝑎 (18)
hfg is steam’s latent heat of vaporization, which is a function of the steam temperature.
ℎ𝑓𝑔 = 𝑓(𝑇𝑠) (19)
26
However, in an ACC, the latent heat rejected during steam condensation is extracted by
the convection heat transfer mode, which is given in the following equation:
= ℎ𝐴∆𝑇 = 휀ℎ𝐴(𝑇𝑠 − 𝑇𝑎𝑖) (20)
The energy balance on the condenser yields:
𝑠ℎ𝑓𝑔 = 휀𝑈𝐴(𝑇𝑠 − 𝑇𝑎𝑖) (21)
And then the steam temperature can be expressed as:
𝑇𝑠 =𝑠ℎ𝑓𝑔 + 𝑈𝐴𝑇𝑎𝑖
𝑈𝐴 (22)
Or
𝑇𝑠 =𝑠𝑓(𝑇𝑠) + ℎ𝐴𝑇𝑎𝑖
ℎ𝐴 (23)
Where UA is the overall heat transfer coefficient between the air and steam. Then from
the definition of the heat exchanger effectiveness:
휀 =𝑎𝐶𝑝𝑎(𝑇𝑎𝑜 − 𝑇𝑎𝑖)
𝑎𝐶𝑝𝑎(𝑇𝑠 − 𝑇𝑎𝑖)= 1 − exp −𝑁𝑇𝑈 (24)
NTU is the number of transfer units defined as:
𝑁𝑇𝑈 =𝑈𝑖𝐴𝑖
𝐶𝑝𝑎𝑎 (25)
Where, Ui is the total thermal resistance between the steam and air, found by equation
(7).
In equations (23) and (24), the only unknown is the air temperature at the outlet.
The cooler area is fixed. The ambient temperature is known from site conditions; the mass
27
flow rate of air and heat transfer coefficient are known from air-side analysis, and the
steam-side heat transfer rate is known from the steam condensing analysis. Then from
equations (18) and (24) the steam temperature can be presented as:
𝑇𝑠 =
𝑠𝑓(𝑇𝑠) + 𝑎𝐶𝑝𝑎𝑇𝑎𝑖
𝑎𝐶𝑝𝑎− 𝑇𝑎𝑖
1 − exp −𝑁𝑇𝑈+ 𝑇𝑎𝑖
(26)
By calculating the condenser steam temperature, the condenser pressure can be
determined from the steam temperature by applying the stated assumption of isothermal
heat transfer under saturation conditions.
Equation (26) shows the high dependency of the steam condenser temperature on
the air side heat transfer and the efficiency of the heat transfer mechanism on it, typically
the fin effectiveness. This study is introducing a novel condenser design using the two-
phase heat transfer mechanism by using the heat pipe as an alternative to the typical
metal fin.
28
CHAPTER 4: PARAMETRIC ANALYSIS OF ANNULAR DISC-SHAPED
HEAT PIPE
In this chapter, the design of an annular disc-shaped heat pipe (ADHP) model is
presented. The general design was introduced in our previous work [31]. In this study, a
detailed analytical study is performed on the designed heat pipe. A one-dimensional
thermal resistance model is derived for investigating the proposed design. In the second
part, the heat transfer limitations of the ADHP are analytically investigated.
Although heat pipes are highly effective thermal conductors, they have heat
transfer limitations. These limitations depend on various design and application
parameters such as the working temperature, the required heat flux, the working fluid,
the type of wick structure. The primary heat transfer limitations are the capillary limit,
which mainly defined by the wick structure and its capability to overcome the pressure
drop over the heat pipe. The maximum heat flux defines the boiling limit that the
evaporator can transfer before the nucleation boiling start to appear on its surface. The
sonic limit that can prevent the vapor from reaching the condenser. The entrainment limit
may occur if the condensed drops block the vapor flow. And the frozen start-up limit
defined by the minimum heat flux required to start the two-phase transition in the heat
pipe [23]. Defining al these limits gives what so-called the heat pipe performance map,
Figure 10 shows a sample heat pipe map, it defines the maximum heat transfer a heat
29
pipe could achieve as a function of its working temperature [22]. In general, in the process
of designing a heat pipe, the working point of the application has to be identified, and
the design should be made to have this working point withing the heat transfer limits
map.
Figure 10: Typical heat pipe performance map [22]
The working fluid and its intrinsic properties strongly influence many of the heat
pipe’s heat transfer limitations. Thus, selecting the proper working fluid type is crucial,
considering the application operating temperature range, the suitability of it with the
wick and solid container materials, and also the availability and financial cost. To avoid
30
the need for extreme vacuum pressure that requires high vacuum equipment and
introduce the risk of high stresses, it is a common practice to select the working fluid to
have its saturation pressure at the required working temperature between 0.1 and 20
bar[23]. Since the operating temperature of the heat pipe under investigation in the scope
of this work is 40-60 °C, and considering its low cost, and the compatibility with the
copper material; water, with a useful range of [30°C – 200°C], is an obvious choice and
was therefore selected for this work.
The focus of the mathematical modeling of the current investigation is primarily
on the effective thermal conductivity and temperature drop of heat pipes. The effective
thermal conductivity is directly related to the heat transfer capability of the heat pipes.
Though the characteristic limitations of heat pipes are of great importance, the resulting
thermal conductivity of the heat pipes takes precedence. Only after the mathematical
modeling is accomplished, the limitations are looked at, specifically the capillary and
boiling limit.
The solution proposed in this study, using an annular disc-shaped heat pipe
condenser, was introduced in our previous work [31]. Figure 11 depicts the proposed
condenser design. On this design, the turbine exhaust steam extracts the heat energy to
the atmosphere air through a series of annular disc-shaped heat pipes staked on the steam
31
pipe outer surface. In this configuration, the evaporator section is the inner wall of the
heat pipe, and the condenser section is the outer surface of the heat pipe.
Figure 11: Proposed Air-cooled condenser design using annular disc-shaped heat pipes
Figure 12 shows the cross-section view of the proposed heat pipe design. The heat
energy extracted from the exhaust steam through the tube wall vaporizes the water in the
evaporator wick region. The generated vapor pressure drives the vapor radially to the
colder region where it condenses on the top and bottom surface of the disc, releasing its
latent heat to the heat pipe container solid wall. The heat is then conducted through the
condenser top and bottom walls to the external cooling air passing in between the staked
heat pipes. The condensed liquid in the top and bottom wick radially flows back to the
32
evaporator side by the act of the wick structure capillary force to passively close the
operation cycle. The gravity forces drive some of the condensed liquid from the top wick
to fall into the bottom side wick. However, the total flow of the liquid should all go back
to the center evaporator. The evaporator surface is also surrounded by a wick structure
to ensure the distribution of the condensed liquid and to use the evaporator surface
effectively.
The design also proposing enhancement of the external air-side heat transfer by
adding dimples features to the outside surface, the dimples increase the convection heat
transfer area between the heat pipe and the external airflow, which increase the
convection heat transfer rate.
The proposed material for the heat pipe external wall is plastic for the advantage
of weight, cost, manufacturing, and maintenance. Even though plastic materials have a
relatively weak conductance, the external wall has a small thickness, which makes the
wall thermal resistance relatively very small in this case.
33
Figure 12: A cross-sectional view of an annular disc-shaped heat pipe [31]
To increase the condensation heat transfer rate on the steam side, the study
proposing using a super-hydrophobic coating on the inner side of the turbine exhaust
tube. The hydrophobic coating prevents the tube’s internal surface from getting wet,
which changes the condensation mode from film to dropwise condensation. This
increases the heat transfer coefficient by order of magnitude, which significantly reduces
the inner tube condensation thermal resistance in comparison to the other thermal
resistances in the thermal path; therefore, reliable correlations for the condensation
process are not needed [55].
As discussed in Chapter 3, the thermal performance of the heat exchanger is
strongly affected by the contact (welding) resistance between the fins and the steam tube
34
external wall. Mainly due to its relatively high thermal resistance and the high cost of
advanced technologies. The proposed design of the heat pipe heat exchanger, as
illustrated in Figure 11, eliminates the contact resistance—the annular disc-shaped heat
pipes made of molded plastic in long arrays. The heat exchanger is assembled by sliding
these arrays over the steam tubes. The temperature deviation along the steam tube creates
a pressure difference between the staked heat pipes, for that, O-rings are used to isolate
the heat pipes to prevent the working fluid from moving in between the ADHP. This
study considers this conceptual design. However, a more detailed study is suggested for
manufacturing considerations such as the selection of container material, wick structure
type, and working fluid. And to consider the assembly of the heat exchanger as a
complete unit. This study provides comprehensive data for the conceptual design
validation that can be used for heat pipe optimization study.
Heat Pipe Thermal Resistance Analysis
The thermal resistance model of the disc-shaped heat pipe is developed to predict
the thermal performance of the proposed design. From the literature, the most
meaningful metric quantifying the heat pipe performance is the effective thermal
resistance or effective thermal conductivity. The presented analysis considers all the
major thermal resistances along the path of the heat transfer in the heat pipe. The
35
performance of a heat pipe operating under the maximum overall heat transfer rate can
be characterized by the total thermal resistance [24, 56-59]. Hence, the heat pipe overall
heat transfer rate can be defined as:
𝑄 =∆𝑇
𝑅𝑡𝑜𝑡 (27)
Where ΔT is the overall temperature difference between evaporator, turbine exhaust
steam in this application, and the external cooling air that is passing over the condenser
surface, and Rtot represents the total thermal resistance between the two temperatures.
Figure 13 illustrates the thermal resistance network for the annular disc-shaped
heat pipe proposed in this study. The figure shows a cross-section view of half of the disc-
shape. The relative sizes of the heat pipe regions in the schematic are adjusted to present
it clearly. In this model, the conduction heat transfer in the radial direction through the
solid container and the wick structure, R10, and R11, respectively, are neglected. This
assumption is reasonable in our design due to the Viton insulation placed between the
evaporator and condenser. And considering the small contact area between the
condenser and evaporator wick.
36
Figure 13: Annular disc-shaped heat pipe thermal resistance map.
The description of each thermal resistance is listed in Table 2, along with its
comparative value for typical heat pipe [60]. The overall thermal resistance is obtained
from:
𝑅𝑡𝑜𝑡 = ∑ 𝑅𝑖 (28)
Ri is the thermal resistance of the heat pipe components and calculated using equations
(29) to (39).
• Inner tube condensing resistance (evaporator heat source)
𝑅1 =1
ℎ𝑒𝐴𝑒 (29)
37
Where h is the convection heat transfer coefficient of the evaporator, and A is the surface
area. The evaporator heat source could be in any form of heat input. For the proposed
annular disc-shaped heat pipe, the evaporator heat source represents the steam
condensing on the inner side of the steam tube.
Table 2: Comparative values for heat pipe thermal resistances
Thermal
Resistance
Description Comparative
values [°C/W]
R1 and R9 Convection between heat source/sink and heat
pipe wall
10-3 to 10+1
R2 and R 8 Evaporator and condenser wall resistance (normal
to the surface)
10-1
R3 and R7 Evaporator and condenser wick resistance (normal
to the surface)
10+1
R4 and R6 Resistance of the liquid-vapor interface. 10-5
R5 Radial resistance of the vapor 10-8
R10 Wick thermal resistance in the radial direction 10+2
R11 Wall thermal resistance in the radial direction 10+4
38
• Tube solid-wall resistance
𝑅2 =ln (
𝑟𝑤
𝑟𝑒)
2𝜋𝑡𝑒𝑣𝑎𝑝𝑘𝑠 (30)
With tevap (=2tv+2tw) is the total height of the evaporator surface exposed to the
wick. And ks is the solid’s thermal conductivity.
• Evaporator saturated-wick resistance
𝑅3 =ln (
𝑟𝑣
𝑟𝑤)
2𝜋𝑡𝑤𝑖𝑐𝑘𝑘𝑒𝑓𝑓 (31)
Where twick is the height of the evaporator wick (twick = tevap). And The effective
thermal conductivity of the saturated wick, keff, depends on the wick type. For the wire
screen mesh, the expression derived by Rayleigh [61] is used, equation (32):
𝑘𝑒𝑓𝑓 =𝑘l[𝑘𝑙 + 𝑘𝑠 − (1 − 휀)(𝑘l − 𝑘𝑠)]
𝑘l + 𝑘𝑠 + (1 − 휀)(𝑘l − 𝑘𝑠) (32)
Where kl and ks are the liquid and solid thermal conductivity, respectively. ε is the
porosity of the wick structure and is calculated by
휀 = 1 −πSNdw
4 (33)
S is the crimping factor (S= 1.05) presented by Chi [62]. N is the mesh number, and dw is
the wire diameter of the screen mesh.
39
And for the metal fiber wick using the expression derived by Mantle and Chang [63]
equation (34)
𝐾𝑒𝑓𝑓 = 𝐾𝑠 [1 +휀
[(1 − 휀)/𝑚] + [𝑘𝑠/(𝑘𝑙 − 𝑘𝑠) ]] (34)
𝑚 = [1.2 − 29 (𝑑
𝑙)] (0.81 − 휀)2 + 1.09 − 2.5 (
𝑑
𝑙) (35)
Up to our knowledge, there is no reported expression to estimate the porosity of fiber
metal wick; however, there are some reported experimental values [63]. In this study, the
value reported by the manufacturer is used.
• Vapor resistance
𝑅5 =𝑇𝑣∆𝑃𝑣
𝑅𝑄𝜌𝑣 (36)
The vapor resistance is dependent on the vapor pressure drop. And as the heat
pipe is working just below the limits; The vapor pressure drop in the heat pipe under
investigation is in the range of 1%. Hence the vapor resistance was neglected in this
analysis [64, 65]
• Condenser saturated-wick resistance
𝑅7B = R7T =𝑡𝑤
𝐴𝑐𝑘𝑒𝑓𝑓 (37)
In Figure 13, the thermal resistances 6 to 9 are split for top and bottom surfaces. Ac
is the condenser top or bottom surface area.
40
• Condenser solid-wall resistance
𝑅8B = R8T =𝑡𝑠
Ac𝑘𝑠 (38)
• Condenser-ambient resistance
𝑅9B = 𝑅9T =1
2ℎ∞𝐴𝑐 (39)
The external heat transfer coefficient is a function of the cooling medium and the
condenser shape. Equation (13) is applicable, assuming the proposed disc-shaped heat
pipes are aligned linearly over a steam tube. However, for our investigation, the heat
transfer coefficient was obtained using a three-dimensional model of the designed heat
pipe inside the wind Tunnel, the heat pipe design and wind tunnel are described in detail
in the Experimental Setup chapter. The problem was simulated using the commercial
CFD software STAR CCM+. The condenser boundary was model with total heat input
equal to the heat input used in the experimental procedure. Figure 14 & Figure 15 show
the condenser external face velocity and heat transfer coefficient under 600 w total heat
input and 42 m/s inlet velocity. With the assumption of symmetry conditions on the top
and bottom condenser surfaces.
There is a stagnation region at the beginning of the heat pipe wall due to the large
thickness of the heat pipe’s sidewall (1.5 in) that facing the airflow. The spots of
41
separation and recirculation at the front side of the heat pipe is very noticeable. The large
insulation cylinders used on the top and bottom caused a high circulation behind it. The
effect is also apparent by looking at the heat transfer coefficient distribution over the
surface. However, by analyzing the affected regions in Figure 15, region (1) is the area
occupied by the insulated sidewall with 1.5 in thickness, and no heat transfer is expected
to be transferred from this region. Region (2) is on the backside of the 2 in insulation
cylinders. And it has a thickness less than the diameter of the cylinders. This penalty is
accepted, as it also increases the heat transfer coefficient on the area further away from
the evaporator centerpiece. Finally, the calculated average heat transfer coefficient values
over the condenser surface are presented for a range of total heat input in Figure 16
Figure 14: CFD velocity distribution over the annular-disc shaped heat pipe
42
Figure 15: CFD - Heat transfer coefficient over the condenser face
Figure 16: Calculated average heat transfer coefficient over the condenser external surface
Note that the resistance values of top and bottom condenser sides are equal under
the assumption of symmetric conditions, which means having an equal temperature of
43
the top and bottom surfaces. By comparing the values in Table 2, the heat conduction
between the solids of the evaporator and condenser was neglected (R11 = 0), same
assumption valid for the conduction between the evaporator and condenser wicks (R10
= 0)
Appling the simplification assumptions stated; using equations (29) - (39) back in
equation (28) the total thermal resistance of the pipe can be written as:
𝑅𝑡𝑜𝑡 = 𝑅1 + 𝑅2 + 𝑅3 + 𝑅𝑐𝑜𝑛𝑑 + 𝑅9 (40)
Rcond is the total resistance of the condenser wick and wall on the top and bottom sides.
1
𝑅𝑐𝑜𝑛𝑑=
1
𝑅7𝑇 + 𝑅8𝑇+
1
𝑅7𝐵 + 𝑅8𝐵 (41)
With the assumption of symmetry conditions on the top and bottom sides:
𝑅𝑐𝑜𝑛𝑑 =𝑅7𝑇 + 𝑅8𝑇
2 (42)
The theoretical overall thermal resistance can be calculated by substituting
equations (30 -(31) & (35 - (39) into equation (40)
in our experimental work, presented in Chapter 5, the temperature measured
inside the centerpiece evaporator cylinder, so R1 is not included in the analysis. And to
investigate the influence of the external heat transfer rate, the heat pipe effective thermal
44
resistance is identified by excluding R1 and R9 from equation (40). As illustrated in Figure
17.
Figure 17: Effective thermal resistance of the ADHP
Heat Transfer Limitations
Although heat pipes devices are known for being very efficient heat transfer
conductors, its performance and operation constrained with several heat transfer
limitations [23]. The heat pipe limitations depend on various parameters, such as size and
shape of the heat pipe components, working fluid type, material and parameters of wick
type, and working fluid [55]. Physical phenomena that may affect the heat pipe’s heat
transfer performance include capillary forces, choked flow, initial boiling, and interfacial
shear [23, 66].
45
Capillary Limitation
The heat pipe performance is solidly dependent on its design shape, wick and case
material, and the type of working fluid. However, its operation is governed by the
capillary pressure difference across the liquid-vapor interfaces in the evaporator and
condenser [24]. The capillary limit is one of the significant parameters that affect heat
pipe performance. Usually, it is a significant heat transfer limiting factor, especially in
low-temperature working heat pipes [26]. The stable working fluid circulation in a heat
pipe is achieved through the capillary pressure head developed by the wick structure.
For properly and steady operation of a heat pipe, the net capillary pressure head must be
higher than the summation of all the pressure losses occurring on the liquid and vapor
flow paths in the wick and vapor regions. The heat pipe pressure balance is expressed as
(Δ𝑃𝑐)𝑚𝑎𝑥 ≥ ∆Pv + ∆Pl + Δ𝑃𝑒,𝛿 + Δ𝑃𝑐,𝛿 + Δ𝑃𝑔 (43)
Where
(Δ𝑃𝑐)𝑚𝑎𝑥 =
Maximum capillary pressure differences the wick structure can
generate.
∆𝑃𝑣 = Total vapor pressure drop occurs in the vapor region
∆𝑃𝑙 = total liquid pressure drop occurs in the wick region
46
Δ𝑃𝑒,𝛿 = The pressure drop across the phase change in the evaporator
Δ𝑃𝑐,𝛿 = The pressure drop across the phase change in the condenser
Δ𝑃𝑔 = Gravitational forces.
Figure 18 shows a schematic of the vapor and liquid pressures along the heat
pipes. The effect of the gravity forces' presence is very noticeable in the liquid pressure.
The gravity pressure may have a positive or a negative sign, depending on the heat pipe
orientation and the evaporator’s location as above or below the condenser. and it can be
expressed as
∆𝑃𝑔 = 𝜌𝑙𝑔𝐿𝑡 sin 𝜑 (44)
Where φ is the heat pipe inclination angle from the horizontal axis; thus, this
pressure drop is negligible when using the heat pipe in a horizontal orientation. The
vapor pressure drop in the condenser and evaporator arises from the friction and inertia
forces, which provide some pressure recovery in the condenser as the mass gradually
reduce through condensation.
47
Figure 18: Pressure variation in a heat pipe
The maximum capillary pressure is derived from the Young-Laplace equation,
which is considered the fundamental equation for capillary pressure [62]
∆𝑃𝑐,𝑚𝑎𝑥 =2𝜎
𝑟𝑒𝑓𝑓 (45)
Where reff is the effective capillary radius. Faghri [23] has proposed empirical
expressions for the wick effective radius and permeability. The expressions for the wick
types used in this study are listed in Table 3. It must be mentioned here that it is
recommended to obtain the values for the wick properties experimentally. However, that
type of experiment is out of this study scope.
48
Table 3: Expressions for wick effective capillary radius and permeability
Screen type Capillary radius Permeability Comments
Wire screen
(Type 1)
1
𝑁 𝑜𝑟
𝑑 + 𝑤
2
𝑑2휀3
122(1 − 휀)2
N = mesh number
d = wire diameter
w = opening size
ε = porosity
Fiber metal
(Type 2)
𝑑
2(1 − 휀)
𝐴(𝑋2 − 1)
𝑋2 + 1
𝑋 = 1 +𝐵𝑑2휀3
(1 − 휀)2
d = fiber radius
A = 6 x 10-10 m2
B = 3.3 x 107 m-2
As been discussed, the capillary limit is constrained by the maximum pumping
capability of the wick structure. For the heat pipe to run under steady-state, it requires a
continuous flow of the condensed flow from the condenser region to the evaporator. At
a specific limit, the evaporation rate exceeds the amount of mass flow return, which
causes an increase in the evaporator region and the dry-out of some spots. This limit
strongly depends on the characteristics of the used wick. i.e., type, porosity, material, and
permeability.
Figure 19 shows the variation of the maximum capillary pressure, permeability,
and effective thermal conductivity with the mesh number of a copper screen wick.
49
Assuming working fluid at 30°C, such as σ = 7.12 x 10-3 N/m, a list of the calculated values
is presented in Table 4.
Table 4:Variation of screen wick properties with its mesh number
Mesh
number
[in]
Porosity
[]
k_eff
[w/m.k]
Permeability
[m2]
R_eff
[m]
Max
capillary
[Pa]
10 7.94E-01 9.41E-01 3.89E-08 1.27E-03 1.13E+02
16 7.62E-01 1.00E+00 1.35E-08 7.94E-04 1.81E+02
20 7.36E-01 1.06E+00 7.75E-09 6.35E-04 2.27E+02
22 7.28E-01 1.08E+00 6.20E-09 5.77E-04 2.49E+02
30 7.03E-01 1.14E+00 3.00E-09 4.23E-04 3.40E+02
40 6.70E-01 1.23E+00 1.46E-09 3.18E-04 4.54E+02
50 6.29E-01 1.35E+00 7.74E-10 2.54E-04 5.67E+02
60 6.29E-01 1.35E+00 5.37E-10 2.12E-04 6.80E+02
80 6.37E-01 1.32E+00 3.14E-10 1.59E-04 9.07E+02
100 6.29E-01 1.35E+00 1.93E-10 1.27E-04 1.13E+03
145 7.37E-01 1.06E+00 1.48E-10 8.76E-05 1.64E+03
50
Figure 19: Variation of Capillary maximum pressure, thermal conductivity, porosity, and
permeability vs. the wick mesh number.
The permeability is a property of the wick material that measures its ability to
transmit the liquid under an applied pressure difference. The liquid flow in the thin wick
region is related to the viscous frictional forces and inertial forces. However, in the
operation of low-temperature heat pipes, the viscous forces are the dominant term, which
is inversely related to the wick permeability. Thus, the higher the permeability, the lower
the pressure drop in the liquid flow. and the flow can be simulated by the Darcy low [62]
𝑑𝑃𝑙
𝑑𝑟=
𝜇𝑙
𝐾𝑢 =
𝜇𝑙𝑚𝑙
𝜌𝑙𝐴𝐾 (46)
51
In the low-temperature heat pipe operation, the mass flow rate can be related to the
amount of heat input by the vapor’s latent heat at the heat pipe working temperature
[62].
=𝑄
ℎ𝑓𝑔 (47)
On the other hand, having high permeability requires a small capillary effective
radius, as can be seen from expressions in Table 3., which means that increasing the wick
permeability must be paid by having a reduction in the wick maximum capillary
pressure.
Figure 20 shows the variation of the permeability, thermal conductivity, and maximum
capillary with the change in the porosity of a copper wire screen mesh. The values
calculated a screen mesh with d=5.59 x10-3 mm, using water at 30°C as a working fluid.
The calculated values are presented in
52
Table 5: Variation of screen wick properties with porosity
Porosity K_eff [w/m.K] Permeability [m2] Max capillary [Pa]
0.05 22.80 3.54E-15 5.87E+03
0.10 11.44 3.16E-14 5.56E+03
0.20 5.50 3.20E-13 4.94E+03
0.30 3.48 1.41E-12 4.33E+03
0.40 2.47 4.55E-12 3.71E+03
0.44 2.19 6.95E-12 3.46E+03
0.60 1.44 3.46E-11 2.47E+03
0.70 1.15 9.75E-11 1.85E+03
0.80 0.93 3.28E-10 1.24E+03
0.90 0.76 1.87E-09 6.18E+02
Figure 20: variation of permeability, thermal conductivity, and maximum capillary pressure
with the porosity of metal screen mesh.
53
In conclusion, it is an optimization process to select the most fitted wick type and
parameters for the heat pipe design and application. In the proposed annular disc-shaped
heat pipe design, the condenser cooling surface is relatively large; and the thermal
conductivity was not an as significant factor in the selection process. And the large vapor
region size gave us the flexibility of using multi-layers of the screen to increase the liquid
flow area without affecting the vapor flow cross-section area. That decreased the viscous
forces acting on the liquid flow without the compromise in the effective capillary radius
of the wick.
The vapor pressure strongly depends on the vapor region design, and its effect on
the total pressure drop is a case to case decision. Ababneh [67] has presented a detailed
model to evaluate the effect of the vapor pressure drop over the total capillary pressure
in a flat heat pipe. His analysis considered four possible models of the pressure drop,
considering all possible acting forces on the model, including frictional, inertial, interface
forces. Applying his model using the designed heat pipe parameters under investigation;
the vapor and inertial forces can be neglected in the model of liquid flow pressure drop
and the overall heat pipe pressure drop for the capillary limit calculations.
54
Boiling Limit
The boiling limit is known as the heat flux limit, as it defines the maximum heat
flux that can be transferred through the evaporator wall. At an adequate heat flux input
to the evaporator wall, the wall temperature becomes excessively high, and the nucleate
boiling starts on the liquid in the evaporator wick. The vapor bubbles that form on the
wall get trapped in the wick and prevent the liquid from flowing back to wet the
evaporator wall, which causes hot spots on the evaporator wall and starts to dry out the
evaporator wick [24]. The boiling limit is defined by the orientation and design of the
evaporator. For conventional cylindrical heat pipe, where the heat input is on the radial
direction normal to the evaporator wall, the boiling limit is the radial heat flux limitation
compared to the axial heat flux limitation for other heat pipe limits [23]. However, the
design of the annular disc-shaped heat pipe under investigation in this study makes the
evaporator heat flux in the same direction as the vapor and liquid flow in the radial
direction, as shown in Figure 12.
The boiling or heat flux limit determination is based on nucleate boiling theory. It
is comprised of two separate phenomena, bubble formation, and the subsequent growth
or collapse of the bubbles [24]. The formation of bubbles is governed by the temperature
difference between the evaporator wall and the working fluid, and the number and size
55
of the formed nucleate sites [23, 24]. Equation (48) defines this critical temperature
difference, in terms of the maximum heat flux.
𝑄𝑚𝑎𝑥 = (𝐾𝑒𝑓𝑓
𝑇𝑤𝑖𝑐𝑘) Δ𝑇𝑐𝑟 (48)
Keff is the effective thermal conductivity of the saturated wick, defined by equation
(32). And ∆Tcr is the critical temperature difference, superheat temperature, defined by
equation (49) [56].
Δ𝑇𝑐𝑟 =T𝑠𝑎𝑡
ℎ𝑓𝑔𝜌𝑣(
2𝜎
𝑟𝑛− ∆𝑃𝑐,𝑚) (49)
Tsat is the saturation temperature of the working fluid. σ is the liquid surface
tension. ∆Pc,m is the maximum capillary pressure difference, which is a function of both
the fluid surface tension and the critical wick radius. And rn is the critical nucleation site
radius, which is assumed to be from 2.54x10-5 m to 2.54x10-7 m [68]. The critical
temperature difference represents the temperature drop across the wick region to
maintain the nucleation bubbles of radius rn at the wall-wick interface.
The behavior of the established vapor bubble on the evaporator surface is
dependent upon the working liquid temperature and the pressure difference across the
liquid-vapor interface caused by the vapor pressure and liquid surface tension. The heat
flux beyond which the bubble growth may happen is developed by using the Clausius-
56
Clapeyron equation to relate the pressure and temperature over the liquid-vapor
interface, to perform the pressure balance on the vapor bubble [62]. The maximum heat
limit for the annular disc-shaped heat pipe under investigation can be determined using
equation (50)
𝑄𝑚𝑎𝑥 =2𝜋𝐾𝑒𝑓𝑓ℎ𝑒𝑣𝑎𝑝𝑇𝑣
ℎ𝑓𝑔𝜌𝑣 ln (𝑟𝑣𝑎𝑝
𝑟𝑤𝑖𝑐𝑘)
(2𝜎
𝑟𝑛− ∆𝑃𝑐,𝑚) (50)
As shown in equation (50), the boiling limit of the heat pipe is highly dependent
on the design shape of the heat pipe, wick material, and the vapor temperature, also
known as the working temperature, which is strongly affected by the condenser side
cooling capacity. In this study, the condenser was cooled by forced convection in the air
tunnel, which simulates the real conditions of the proposed heat pipe application in air-
cooled condensers. The experimental setup is explained in Chapter 5. This setup limits
the capability to investigate the boiling limit by having a relatively small heat transfer
coefficient on the condenser side. Typically, to measure the boiling limit, one needs to
increase the heat input to the evaporator while keeping the vapor temperature constant
by increasing the cooling capacity in the condenser side, by increasing the cooling
medium flowrate. Figure 21 illustrates the effect of the nucleate radius value on the
variation of boiling limit, represented by the maximum heat input, with the vapor
temperature. The values were calculated using equation (50) for the annular disc-shaped
57
heat pipe design presented in Figure 13 using the parameters listed in Table 6, with
thermal properties calculated at the vapor temperature.
Figure 21: effect of vapor temperature and nucleation radius on the boiling limit
Table 6: Parameter values for the boiling limit.
Parameter Unit Value
Keff [W/cm. K] 2.100 E-2
hevap [cm] 3.175
rvapor [cm] 3.226
rwick [cm] 3.175
Mesh # (N) [cm-1] 57.1
58
CHAPTER 5: EXPERIMENTAL SETUP
an experimental study was designed, constructed, and conducted to investigate
the characteristical thermal performance of the designed annular disk-shaped heat pipe.
This chapter presents the process and stages of the experimental study through all the
designing, building, and conducting phases. The main aim of the study to reduce the air-
side thermal resistance of the steam condenser; the experiment is designed to measure
the total thermal resistance of the heat pipe under various configurations. However, due
to the cost and time required for building the heat pipes, a benchmark case is designed
and conducted. The results of the experimental data are used to validate the theoretical
model that is used for further investigation of the ADHP. The main design drive of the
heat pipe shape is its proposed application that requires it to extract the heat from the out
surface of the steam tube. A circular disc-shaped is selected considering its low-cost
manufacturing and processing. However, the idea of the annular heat pipe could be in
different shapes, such as elliptic or more flattened shapes, to adapt to the steam condenser
tubes. The heat pipe was extensively tested under transient and steady-state conditions.
The heat pipe was air-cooled using a variable-speed blower that gave the flexibility to
test the heat pipe under different airspeed conditions.
59
Annular Disc Heat Pipe Design
The disc-shaped heat pipe was designed for the steam condenser application. In
the experimental study, the heat transfer from the steam side was simulated by adding a
heat source to the disc’s inner surface. The heat pipe was air-cooled to test the design
under similar real application conditions. However, this cooling method came in a
compromise of the total heat transfer amount through the condenser. Hence, a large
condenser surface area required to achieve considerable total heat transfer. The wind
tunnel size also constrained the design of the condenser size. This study did not cover the
optimization of the heat pipe surface. However, the results obtained establishes an
adequate base for an optimization study for future work and development.
Heat Pipe Design Considerations
Considering the proposed application purpose of the disc-shaped heat pipe, it did
not feature any adiabatic section. In the meaning of heat pipe adiabatic section purpose.
The heat pipe was designed in the CATER lab, considering the below main features to
allow for variation and freedom in the experiment matrix based on results analysis:
1- Flexibility of vacuuming the system to any absolute pressure.
2- Flexibility of charging the system initially and changing the charge ratio.
60
3- Simple assembly procedure.
4- The capability of using different types and configurations of wick structures.
5- The ability to conduct the experiment with or without the adiabatic section.
6- The ability to measure the transient internal vapor temperature and pressure.
The ability to run the heat pipe under different initial vacuum pressure gave us an
understanding of the effect of the existence of non-condensable gases on the heat pipe
performance. The second feature was considered to study the effect of the working fluid
charging amount on the heat pipe performance and to identify the range of the optimum
filling ratio. The wick material and structure have a significant effect on the heat pipe
performance and heat transfer limitations, so it was essential to consider the simplicity of
opening and re-assembling the heat pipe with the ability to change the wick type or
thickness.
Annular Disc-Shaped Heat Pipe Dimensions
Figure 22 shows a cutaway view of the annular disc-shaped heat pipe. The heat
pipe consists of two 110 copper plates. The plate’s dimensions are 1/8 in thickness, 2 in
inner diameter, and 10.4 in outer diameter, as shown in Figure 23. The plates are
sandwiched over a 1.25” thick aluminum tube with 9 in and 10.5 in inner and outer
61
diameters, respectively, as shown in Figure 24. The aluminum tube (sidewall) surface was
anodized with a black hard-coat finish to eliminate any chemical reaction with the
working fluid (water). Considering the thickness and surface coating of the sidewall, the
heat transfer on the sidewall was neglected. The top and bottom plates were mounted to
the sidewall with 24 of 1/4 -20 size bolts on each side, to minimize the heat transfer
through the bolts Teflon washers were used to seal and lock the bolts to the copper
surface. The surface connection between the condenser plates and the sidewall was sealed
by a 2-271 O-ring placed in a grove located between the bolts and the tube inner wall
edge.
Figure 22: Cutaway view of the annular disc-shaped pipe
62
Figure 23: Condenser copper plate dimensions
Figure 24: Sidewall of the annular disc-shaped heat pipe
63
A 1.5 in height and 2.5 in diameter cylindrical 110 copper block was also
sandwiched between the two copper discs. The radial outer surface of the cylindrical
center copper block was considered as the evaporator section, and the outside surface of
two copper disc plates are the condenser surface. The two condenser copper plates,
aluminum sidewall, and center copper piece were aligned to be concentric. Four 150 w
cartridge heaters with 0.5 in diameter and 1 in length were inserted in the copper
centerpiece in symmetrical position patterned around the perimeter of 1.26 in circle, as
shown in Figure 25. Two 2 in thick Rohacell cylinders were used to insulate the top and
bottom surface of the evaporator.
Figure 25: Dimensions of the center copper cylinder (evaporator)
To ensure the alignment of the condenser plates, evaporator block, and the
sidewall tube, the sidewall outside diameter was 0.1 in bigger than the condenser top and
64
bottom plates. A tip of 0.05 in thickness of the aluminum sidewall outer diameter was
extracted vertically for 0.125 in on both sides (up and down), to lock the condenser copper
plates in concentric location mechanically. The condenser plates lock into the evaporator
copper block over 0.25 extended step with 1/8 in depth, as shown in Figure 26. Two Viton
gaskets were placed between the evaporator and condenser plates to seal the connection
in between and to increase the thermal contact resistance to minimize the conduction heat
transfer between the evaporator and condenser. The size of the gaskets used was 2 in, 2.5
in, and 0.031 in inner diameter, outer diameters, and thickness, respectively. Figure 27
shows the built annular disc-shaped heat pipe without the copper top plate to show the
internal components of the assembly.
Figure 26: Mechanical lock assembly of the heat pipe components
65
Figure 27: Annular disc-shaped heat pipe with an open-top
Three different types of wick material were tested, two copper screen mesh wicks
with 145 and 200 mesh numbers, and an AISI 316L stainless steel fiber structure wick.
The wick material was wrapped around the evaporator block and hold in place using two
thin copper wires located at approximately 0.5 in from the evaporator top and bottom
sides. Six acrylic posts (0.5 in x 0.5 in) were used to hold the layers of wick against the
inner faces of the top and bottom condenser copper plates. The specifications of the used
wick types are listed in Table 7. The effective thermal conductivity of the wick was
66
calculated based on the working fluid of water. The built heat pipe dimensions and
materials are presented in Figure 28, Figure 29, and
Table 8
Table 7: wick specifications
Wick #1 Wick #2 Wick #3
Type Screen Screen Fiber
Material Copper Copper SS 316
Mesh number [in-1] (m) 145 (3683) 200 (5080) -
Porosity 46% 35% 60%
Permeability [m2] 8.54x10-8 2.15x10-8 2x10-8
Layer thickness [m] 0.11x10-3 0.1x10-3 0.1x10-3
Effective conductivity [W/m/K] 2.07 2.9 0.46
67
Figure 28: Built heat pipe CAD
Figure 29: Cross-section A-A from Figure 28
68
Table 8: Material of the built ADHP
Part Material
Evaporator 110 Cooper
Condenser 110 Copper
Sidewall Anodized Aluminum (black hard coat surface)
Insulation Rohacell
Working Fluid De-Ionized Water
Experimental Measurements
The most common and well-referenced method to investigate the heat pipe
thermal performance is by identifying the heat pipe effective thermal resistance Reff or the
effective thermal conductance. Also, it is very informative to investigate the heat pipe
maximum heat transfer limit Qmax [23-26, 58, 59, 62, 65, 69-71]. The heat pipe thermal
performance was investigated under steady and transient conditions. A data acquisition
system was used to record all the readings of the thermocouples, pressure transducer,
and the heaters voltages. The data acquisition system consisted of a FLUKE 2688A data
logging system and three FLUKE 2680 precision analog input modules to continuously
read the TCs, heaters, and pressure transducer. Each module has a capacity of 20
69
channels. To capture the transient heating process during the test, the DAQ was sat to
read 1 s-1 frequency. A skitch of the experimental setup is presented in Figure 37. Where
(θ = 0) is the radial axis in parallel with airflow. And (h) is the vertical height.
Temperatures
The internal evaporator temperature was recorded using ten T-type
thermocouples distributed along the perimeter of 1.88 in diameter circle. Figure 30 shows
the detailed location of the evaporator thermocouples. The thermocouples inserted in
1/16 in diameter holes. The 10 thermocouples were placed at different depths to check the
heat distribution in the evaporator. Eight thermocouples were placed at 0.75 in depth,
one at 0.5 in depth (TC# 2), and one at 1 in depth (TC# 10). Placing the eight
thermocouples at 0.75 in depth makes it at the center height of the vapor region between
the condenser plates. The thermocouples were placed inside the copper block to avoid
disturbing the vapor flow inside the heat pipe by routing the wires in the vapor region.
The distance between the evaporator surface and the location of the thermocouples was
considered in the calculations assuming one-dimensional heat conduction in the radial
direction by neglecting the axial heat transfer considering the insulation on the top and
bottom sides and the high heat transfer rate at the evaporator surface. The orientation of
70
the evaporator was locked in position, making TC #1 at the center of the evaporator
section facing the cooling wind direction inside the wind tunnel.
Figure 30: Location of the evaporator thermocouples
A total of 28 T-type thermocouples were used to measure the temperature
variation on the condenser outer surface. The thermocouples were distributed along the
radial axis over the top and bottom copper plates of the condenser. The locations of the
temperature thermocouples on one of the condenser’s surfaces are depicted in Figure 31.
As illustrated, the temperature variation was reordered at two angular axes locations,
parallel (θ = 0) and perpendicular (θ = 90°) to the outside wind tunnel cooling airflow.
71
Figure 31: locations of the condenser surface thermocouples
A T-type thermocouple probe was used to monitor the vapor temperature inside
the heat pipe. The thermocouple was inserted from the side wall at (θ = 0) through a ¼ in
NPT tap, as shown in Figure 32. The exact location of the sidewall tap is illustrated in
section A-A in Figure 24. The vapor thermocouple position was fixed at approximately
the center point of the vapor region (r = 3 in and z = 0.75 in), at this location, the vapor
thermocouple is parallel to the condenser surface thermocouple #5 in Figure 31. The
72
change in the working fluid temperature over the vapor region is negligible, and the
vapor region can be considered isothermal [72-74]. To validate this assumption and to
study the vapor temperature variation along the radial axis, one case was repeated with
fixing all the conditions and changing the vapor thermocouple locations (R= 1.5 in, R= 2.5
in, and R= 3.5 in). The results showed a maximum difference of ±1.3°C during the steady
run, which falls within the thermocouple accuracy. Hence the isothermal vapor region
assumption is valid in our case.
The thermocouples were calibrated against an RTD thermocouple with ±0.1°C
accuracy. A separate fitting curve was generated for each thermocouple using the
readings over 4 different temperatures in the range of 5°C – 160°C. In the calibration
process, the maximum allowed temperature difference to the RTD was 1.5°C. The
thermocouples that had higher error were replaced with new ones.
Finally, two T-type thermocouples were used to measure the air temperature at
the inlet of the wind tunnel.
Pressure
The absolute pressure inside the heat pipe was monitored using a pressure
transducer of Omega PX209-015A5V powered by a 5V VDC power supply. The
transducer has a voltage reading accuracy of 0.25% (Linearity, Hysteresis, and
73
Repeatability). The Omega transducer was calibrated against an MKS transducer with
0.25% accuracy, to ensure the accuracy of the vapor pressure application. The calibration
was done by measuring the pressure in a vapor chamber under vacuum was and
increasing the pressure by heating the water inside the chamber. The measured data were
fitted within 0.9% of the fitted linear equation. The calibration process described in more
detail in [75, 76] The pressure transducer was inserted through the heat pipe side wall
across from the vapor temperature thermocouple at θ = 180°. The location of the vapor
pressure transducer is illustrated in Figure 32
Figure 32: location of vapor temperature and pressure measurements
74
Table 9: Pressure transducer calibration measured values
Omega [Volt] MKS [kPa] Omega [KPa) Error [%]
0.178 2.863 2.846 0.569
0.178 2.840 2.865 0.873
0.216 3.610 3.634 0.654
0.228 3.884 3.890 0.149
0.247 4.281 4.278 0.070
0.285 5.072 5.063 0.161
0.301 5.403 5.396 0.128
0.347 6.347 6.345 0.018
0.436 8.185 8.185 0.003
0.456 8.608 8.605 0.032
0.475 9.014 9.011 0.037
0.563 10.811 10.818 0.061
Figure 33: Pressure transducer calibration curve vs. MKS transducer
75
Heaters
Four 1.0 in long AC insertion heaters were used to generate the required power as
the heat source for the evaporator section. The specifications of the used heaters are listed
in Table 10. Important to mention that the heater resistance listed in the table is the value
listed in the heater specification. However, we have measured the exact resistance of each
heater during the experiments, and the exact value was within ±1.5 Ω of the manufacture
specification. The heaters were controlled using a 120 v, 20 amp VARIAC. And each
heater was connected to a rheostat for fine-tuning of voltage. The maximum total heat
input (𝑄𝑡𝑜𝑡𝑎𝑙) achieved in this test was 600 w. where heat input from each heater (𝑄𝑖) was
calculated using the measured voltage (𝑉𝑖) and resistance (𝑅𝑖).
𝑄𝑖 =𝑉𝑖
2
𝑅𝑖 (51)
𝑄𝑡𝑜𝑡𝑎𝑙 = ∑ 𝑄𝑖 (52)
76
Table 10: Insertion heaters specifications
Specification Unit Value
Power Watt [w] 150
Watt density Watt [w/in2] 226
Voltage (AC) Volt [V] 120
Resistance Ohm [Ω] 96
Heating element diameter Inch [in] 0.495
Heating element length Inch [in] 1.0
heated length Inch [in] 0.5
Experimental Heat Loss
The experimental heat losses are identified as the amount of heat that does not
transfer radially from the evaporator through the vapor. The heat losses were measured
by measuring the heat transfer through the insulation cylinders. A total of four T-type
thermocouples were used to measure the temperature in the top and bottom insulation
Rohacel cylinders. Two 1/16 in diameter and 1 in depth holes were drilled at the radial
out surface of each cylinder. By that, the thermocouple locations were along the center
77
axis of the insulation cylinders, as illustrated in Figure 34. The heat loss in the system was
calculated as:
𝑄𝑙𝑜𝑠𝑠% =𝑄𝑙𝑜𝑠𝑠
𝑄𝑖𝑛∗ 100 (53)
𝑄𝑙𝑜𝑠𝑠 =𝐾𝑖𝑛𝑠𝐴𝑖𝑛𝑠∆𝑇
𝐿 (54)
Where Qloss and Qin are the heat loss and total heat input, respectively, Kins is the
thermal conductivity of the Rohacel (= 0.03 w/m.k), Ains is the circular cross-sectional area,
and L is the vertical distance between the thermocouples. The heat losses in the system
were calculated using the measured insulation temperatures in equations (53) and (54),
the maximum calculated heat losses were less than 0.2% of total heat input. Thus, the heat
loss to the ambient was negligible compared to the total heat input, especially at power
input over 300 w, as the loss percentage of 0.1%. Figure 35 shows the calculated heat loss
percentage and temperature difference in the insulation for one experimental case with
heat input up to 700 w.
78
Figure 34: location of temperature measurement in the insulation cylinders.
Figure 35: Variation of heat loss percentage with total heat input
79
Vacuum and Charging Station
The vacuum and charging connections were completed at separate locations using
the taps on the sidewall located at θ= 90° and θ= 270°, respectively. Figure 36 shows the
schematic of the charging station. The heat pipe was vacuumed using Hitachi direct drive
rotary vacuum pump type 160VP with a minimum absolute pressure of 10-1 Pa. A 1/4
NPT compact backflow-prevention valve was used to connect the heat pipe to the
vacuum pump with a pressure limit of 5000 psi. The simple-type charging station, Figure
36, was used to charge the working fluid into the heat pipe, the method has a direct
connection between the working fluid container and the heat pipe [23, 24, 67, 70]. A
Swagelok gate valve was used in between the working fluid (de-ionized water) container
(100 ml graded beaker) and the heat pipe, to control the filling process of the heat pipe.
The beaker was mounted 5 in vertically over the heat pipe filling point. By gradually
opening the valve after vacuuming the heat pipe, the working fluid flows into the heat
pipe by the pressure difference force and by the gravity force as the fluid container placed
above the heat pipe.
80
Figure 36: schematic of the vacuuming and charging station
The system was first vacuumed until the pressure inside does not change for 30
min. After turning off the pump, the gate valve to the filling container was opened and
closed quickly to allow a minimal amount of water into the heat pipe, typically a small
enough amount that it evaporates. Then the vacuum pump started again to pull out the
entrained water along with the non-condensable gases and air remaining in the system.
This procedure was repeated for few times to remove the air and non-condensable gases
until a stable pressure inside the heat pipe was achieved.
81
Test Section
The heat pipe was tested inside the vertical center of a wind tunnel with 6.5 in x
13 in the cross-sectional area. The flow is guided through a flow conditioning section with
5 layers of fine mesh screen in the front, to make sure that the airflow enters the test
section was as uniform as possible.
Figure 37: CAD of the wind tunnel and test section
The boundary condition of the airflow at the test section inlet was defined by
measuring the dynamic pressure using pressure pitot over the flow cross-section area at
82
20cm from the entry of the rectangle test section, as shown in Figure 37. The dynamic
pressure data were used to calculate the mass flow rate and average velocity of the flow.
First, the velocity was obtained from the measured dynamic pressure, ∆P, by:
U𝑖 = √2∆𝑝
𝜌 (55)
Where ρ is the air density calculated by the ideal gas low at the measured temperature
assuming constant temperature over the cross-sectional area. And the mass flow rate
through each cell was calculated by:
𝑖 = 𝜌𝑈𝑖𝐴𝑖 (56)
The sum of all the cells calculated the total air mass flow rate,
𝑡 = ∑ 𝑖 (57)
Then, the average velocity upstream was calculated from:
=𝑡𝑜𝑡𝑎𝑙
𝜌 ∗ 𝐴𝑡𝑜𝑡𝑎𝑙 (58)
The measured values and conditions are described in Appendix A. with the fan running
at 1428 rpm, the average velocity calculated to be 41.96 m/s.
Test Procedure
In general, the heat pipe performance was tested under different configurations of
changing wick type, filling ratio, condenser size, and heat input. To identify the effective
83
thermal conductivity or effective thermal resistance of the designed heat pipe. The
general procedure for all test cases was as follow:
1. Vacuum the heat pipe.
2. Open the filling gate valve and fill the system to the required charge
amount.
3. Start the wind tunnel and wait for a steady flow to be achieved.
4. Start the DAQ system.
5. Start the heaters with a total power of 40w.
6. Monitor the average temperature of the evaporator, condenser, and vapor
to reach steady. (1 min with less than 0.1 temperature variation).
7. Increase the total power by 40 w. and repeat step #6.
8. Repeat step #7 till reaching the maximum total power from the heaters (600
w) or the maximum heat pipe heat limit.
9. Turn off the heaters and monitor all temperatures to reach the ambient
temperature.
Data Reduction
All the temperature, pressure, and voltage readings were continuously recorded
for the experimental case through all the test steps. The measured data were used to
84
evaluate the equivalent thermal resistance and effective thermal conductivity of the heat
pipe as:
𝑅𝐻𝑃 =𝑇𝑒 − 𝑇𝑐
𝑄𝑖𝑛 (59)
Where RHP is the total heat pipe equivalent of thermal resistance, considering the
temperature difference between the condenser and evaporator surfaces. Te and Tc are the
average evaporator and condenser temperatures, respectively.
𝑇𝑒 =∑ 𝑇𝑖
𝑁 (60)
Where Ti is the temperature reading of thermocouples distributed inside the
evaporator copper block, the top and bottom condenser temperatures were measured on
two axes (θ = 0, and θ=90°) with 5 thermocouples on each line on the top plate and 7
thermocouples on each line for the bottom plate, gives a total of four condenser average
temperatures (Tt,0, Tt,90, Tb,0, Tb,90).
𝑇𝑗,𝜃 =∑ 𝑇𝑗,𝜃,𝑖𝑖
𝑁 (61)
Where j=t or b, for top and bottom, respectively. N is the number of used
thermocouples along the axis. The total average of each plate is referred to as Tt and Tb
for the top and bottom condenser side, respectively. And Tc is the average of the 28
thermocouples distributed on the two sides of the condenser.
85
The total thermal resistance of the system considering the external condenser
conviction resistance is calculated as:
𝑅𝐻𝑃𝑇 =𝑇𝑒 − 𝑇∞
𝑄𝑖𝑛 (62)
Uncertainty Analysis
For proper understanding and interpretation of the experimental data, a proper
uncertainty analysis is required to increase the reliability of the experimental results. The
uncertainty analysis consists of two types of error, random and systematic errors. The
systematic error comes from the instruments used in the experiment, and usually comes
due to the wrong calibration or zeroing of the device. And typically, it is repeated and
carried over for each repetition of the experiment. Unlike the random error, that tends to
be normally distributed around a mean. Random errors come from the randomness of
the experiment, human error, or the accuracy of the used devices, which is usually
provided by the device manufacturer. The random error can be reduced to a certain
confidence level by the repetitive test [77] For this study, a confident interval of 95% is
used.
The uncertainty associated with the temperature reading using thermocouples
and DAQ equipment is ± 0.5 K. The uncertainty related to the power supplied to the
86
heaters is approximately ± 5%. Pressure measurement readings have an uncertainty of
4%. Using the methodology proposed by Moffat [78], the error calculated using the model
of equations (63 - (66) accounts for the systematic and random errors. A detailed
explanation of the model is described by Otto [79].
𝑋𝑖 = 𝑋𝑖(𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑) ∓ 𝛿𝑥𝑖 (63)
𝑅 = (𝑋1, 𝑋2, … 𝑋𝑛) (64)
𝛿𝑅𝑥𝑖 =𝜕𝑅
𝜕𝑋𝑖𝛿𝑋𝑖 (65)
𝛿𝑅 = √∑ (𝜕𝑅
𝜕𝑋𝑖𝛿𝑋𝑖)
𝑁
𝑖=1
= √(𝜕𝑅
𝜕𝑋1𝛿𝑋1)
2
+ (𝜕𝑅
𝜕𝑋2𝛿𝑋2)
2
+ ⋯ + (𝜕𝑅
𝜕𝑋𝑁𝛿𝑋𝑁)
2
(66)
Where Xi is the mean of a measurable property, and δXi is the uncertainty
associated with the measurement. R is a result of data processing or a function that
depends on several measurements. So, equation (63) shows the dependency of the
resulting uncertainty on the error of each measurement used in its calculation. And the
partial derivative in equation (65) represents the sensitivity of the result relative to each
measurement. Finally, the total uncertainty of a result is calculated by the root sum square
of all contributing measurements as shown in equation (66)
87
The main processed results in this study are power input as a function of measured
voltage and electrical resistance, the average temperature of evaporator and condenser,
the condenser heat transfer coefficient, and the effective thermal resistance of the heat
pipe as a function of heat input and temperature difference.
Table 11: Experimental uncertainty values
Parameter Uncertainty
Power Input ±1.8 %
Temperatures ±2 oC
Te ±0.7 oC ~ ±1 oC
Tc ±0.41 oC ~ ±1 oC
dT ±0.8 oC ~ ±1.4 oC
R
± 1.82 % dT(min)
± 1.80 % dT (max)
88
CHAPTER 6: RESULTS
Experimental Results
The heat pipe was tested under different configurations. The performance of the
heat pipe was recorded in transient and steady-state for each total power input starting
from 40 w to the maximum of 600 w, with 40 w steps. Each configuration was tested
under different filling ratios of 60 ml, 90 ml, and 120 ml, referred to as A, B, and C,
respectively. The experimental cases are listed in Table 12. The specifications of the wick
types are listed in Table 7. Also, to understand the heat pipe response time during startup
heating and cooling down, it was tested under particular total heat input for 15 min of a
continuous run before cooling down back to the starting conditions of temperature and
pressure.
Table 12: configuration of the experimental cases.
Case # HP
container
Wick
type
No. of top
wick layers
No. of bottom
Wick layers
Filling Amount
0 Copper - 0 0 Empty
1 Acrylic 1 0 5 B
2 Copper 1 5 5 A, B, C,
3 Copper 2 0 5 A, B, C,
4 Copper 3 0 5 A, B, C,
Symbols A = 60 ml, B = 90 ml, C = 120 ml
Wick
types
Type 1 = copper screen mesh number 145
Type 2 = copper screen mesh number 200
Type 3 = stainless steel fiber wire
89
Repeatedly Test
Following the uncertainty discussion in Chapter 5, the repeatedly of the testing
reduces the random error percentage in the measurements. Few experimental cases were
repeated to achieve a reliable level of confidence in the measuring system. As a sample
of the repeatedly comparison test, Figure 38 compares the transient average evaporator
temperature for Case 2C (Table 12) configuration over the run of 15 min at two total
power inputs of 280 w and 600 w. the figure shows a very good repeatedly of the
measurements. Figure 39 shows the absolute pressure transient reading inside the
annular disc-shaped heat pipe for the same case 2C at 600 w.
Figure 38: Transient average evaporator temperature for 120 ml water filling with 5 layers of
145 copper screen mesh (Case 2C)
90
The two figures show the response of the heat pipe temperatures and pressure to
the heating processes during the startup and cooling down. During the startup heating
process, the temperature response is faster than the pressure. The temperature takes
about 3 min to reach the steady-state while the pressure took about 7 min to reach the
flat, steady-state reading.
Figure 39: Transient absolute vapor pressure for 120 ml water filling with 5 layers of 145 copper
screen mesh at 600 w (Case 2C)
To validate the repeatedly of the experimental setup, Case 2B was repeated for
three runs. With disassembling and cleaning the heat pipe for each pipe run before
assembling and charging it again to mitigate any systematic or human error during the
91
assembly. The variation of the vapor temperature and average evaporator temperature
with the heat input are presented in Figure 40 and Figure 41, respectively. The deviation
between the three runs was less than 8%.
Figure 40: Repeatedly test - variation of vaportemperature with heat input for 90ml water with
with 5 layers of 145 copper screen mesh (Case 2B).
92
Figure 41: Repeatedly test - variation of average evaporator temperature with heat input
for 90ml water with with 5 layers of 145 copper screen mesh (Case 2B).
Model and Experimental Validation
To validate the experimental setup and to assure the validity of the data obtained
in this study, the results must be compared to a previously established experimental data.
In this study, the data were compared to the experimental results of North and Avedisian
[80]. Their data were used by Zhu and Vafai [74] to validate their analytical model of an
asymmetrical disc-shaped heat pipe. North has presented two experimental results for
two different heat pipe designs: air-cooled manifold heat pipe and water-cooled disc-
shaped heat pipe. The results showed typical behavior of the heat pipe performance. The
93
analysis presented in the study by North suggested that the performance curve
(evaporator heat flux variation with its temperature) is split into two regions: a single-
phase regime and a boiling or tow-phase regime. The single-phase regime is the starting
region, where conduction is the primary dominating heat transfer mode. The second
regime is where the evaporation dominates the heat dissipation process, and that is
validated by the increase in the heat transfer rate with temperature variation[80]. This
performance analysis applies to any heat pipe shape[80]. For this study, we have used
the performance curve of the disc-shaped heat pipe presented by North for the
comparison; it was selected because of the similarity in the mass and heat dissipation
flow direction.
Figure 42 shows the performance curve for Case 2C and Case 3C, along with the
data presented by North for 116 ml methanol filling. The tested copper disc-shaped heat
pipe by North has an outer diameter of 14.9 cm and 2.54 cm thickness. The heat pipe has
a complete disc-shape with a plain solid bottom and top plates. The evaporator section is
part of the bottom plate with 7.62 cm. Which is different than the proposed design in this
study, in which the evaporator section is the concentric part of the disc. However, as the
primary heat transfer mode in heat pipes is the phase change and vapor flow, the two
heat pipes should have similar performance curves. Thus, the comparison provides
meaningful information for the validation purpose of the experimental setup. Another
94
main difference between the two experiments is that North’s heat pipe was water-cooled
comparing to the air-cooled heat pipe in this study.
Figure 42: Experimental validation of the current study with North [80].
The two cases that are presented in the comparison have similar performance
curves with two identified regimes. The single-phase region of the current study (Ac) has
lower heat dissipation per temperature increase comparing to the same region of North
(AN). This deviation is expected as the conduction is the primary mode in this region. In
the current design, the evaporator is insulated from the condenser by Viton gasket, which
limits the heat transfer and causes the temperature increase in the evaporator. It is also
noted that the boiling region in North (BN) starts at lower temperatures comparing to the
95
same region in the current study (Bc). The differences in the starting pressure explain this
behavior; the lower the starting pressure, the lower the phase change starts. The heat
dissipation in BN region is also higher than Bc. Many factors affect this slope, but the main
factor would be the external condenser heat transfer coefficient. The results reported by
North in [81] for the air-cooled manifold heat pipe showed that the higher airflow rate
yielded lower temperatures and higher heat fluxes than the lower flow rate for the same
heat pipe. And that explains the shifting of the BC slope trend line to left from BN. North’s
heat pipe used a 3 mm thick sintered powder copper wick on each of the disc’s inner
sides. With an effective pore radius of 36 µm, 50% porosity, and 2.1 x 10-11 m2 compared
to the wick properties used in this study listed in Chapter 5. However, these parameters
are expected to affect the maximum heat transfer and capabilities of the heat pipe more
than the steady performance. Figure 43 depicts the performance curve of the empty case
in this study (Case 0 in Table 12 ) comparing to the empty case presented in North’s study.
The curves’ linear slopes are almost identical to the slope of the single-phase region of
each study, which supports the explanation provided above of the dominating of
conduction mode during the starting regime.
96
Figure 43: Performance curve of the empty heat pipe for the current study and North [1]
In the second part of the experimental validation, the experimental condenser
temperature measured at 16 points on the surface was compared to an analytical solution.
The experimental case used for this comparison is the empty heat pipe case (no working
fluid charged). As the heat pipe was vacuumed, the heat transfer through the evaporator
surface is neglected. And as the evaporator top and bottom sides are insulated, the
dominant heat transfer path was the conduction through the top and bottom copper
plates. Either top or bottom plate can be modeled as an annular disc-shaped fin with a
thickness equal to the condenser plate, with constant heat flux at the inner radius of the
97
fin, assuming the symmetry of the two plates. Figure 44 shows a schematic of the cross-
section of the analyzed condenser plate.
Figure 44: schematic of the fin analysis - top condenser cross-section
The heat flux at the fin inner circle is equal to half the measured actual total
evaporator heat input after excluding the losses and divided by the fin cross-sectional
area at the inner radius. The fin geometry is a circular disc with an inner and outer radius
as of the condenser plate dimensions illustrated in Figure 23. The governing equation for
the radial fin is obtained by applying energy balance on the fin:
𝑞(2𝜋𝑟)(𝑡) − 𝑞(2𝜋)(𝑟 + ∆𝑟)(𝑡) − ℎ∞(2𝜋𝑟∆𝑟)(𝑇 − 𝑇∞) = 0 (67)
Rearranging equation (67) by dividing it by (2π∆r), letting ∆r→0, and substituting
Fourier’s law
𝑑
𝑑𝑟(𝑟
𝑑𝑇
𝑑𝑟) −
ℎ∞𝑟
𝑡𝑘(𝑇 − 𝑇∞) = 0 (68)
98
Introducing:
𝛽2 =ℎ∞
𝑡𝑘, 𝑍 = 𝛽𝑟, 𝜃 =
𝑇 − 𝑇∞
𝑇0 − 𝑇∞
Where T0 is the fin temperature at the inner surface. Then equation (68) can be written as
𝑧2𝑑2𝜃
𝑑𝑧+ 𝑧
𝑑𝜃
𝑑𝑧− 𝛽2𝑧2𝜃 = 0 (69)
Equation (69) is a modified Bessel’s equation of zero-order, and the solution for it is
𝜃 = 𝐶1𝐼0(𝑧) + 𝐶2𝐾0(𝑧) (70)
Neglecting the heat transfer through the fin tip, the boundary conditions of the fin are
𝑟 = 𝑟𝑖𝑛 : 𝑑𝑇
𝑑𝑟=
−𝑞𝑖𝑛
𝑘 𝑜𝑟 𝑧𝑖𝑛 = 𝑟𝑖𝑛𝛽:
𝑑𝜃
𝑑𝑧=
−𝑞𝑖𝑛
𝑘𝛽(𝑇0 − 𝑇) (71)
𝑟 = 𝑟𝑜𝑢𝑡 : 𝑑𝑇
𝑑𝑟= 0 𝑜𝑟 𝑧𝑜𝑢𝑡 = 𝑟𝑜𝑢𝑡𝛽:
𝑑𝜃
𝑑𝑧= 0 (72)
Solving equation (70) for C1 and C2 using the boundary condition in equations (71) & (72)
𝐶1 =𝑆𝐾1(𝑧𝑜𝑢𝑡)
𝐹(𝑧𝑖𝑛, 𝑧𝑜𝑢𝑡), 𝐶2 =
𝑆𝐼1(𝑧𝑜𝑢𝑡)
𝐹(𝑧𝑖𝑛, 𝑧𝑜𝑢𝑡) (73)
Where
𝑆 =−𝑞𝑖𝑛
𝑘𝛽(𝑇0 − 𝑇), 𝐹(𝑧𝑖𝑛, 𝑧𝑜𝑢𝑡) = 𝐼0(𝑧𝑖𝑛)𝐾1(𝑧𝑜𝑢𝑡) − 𝐼1(𝑧𝑜𝑢𝑡)𝐾0(𝑧𝑖𝑛) (74)
The analytical solution is compared to the experimental results of the evacuated
run case. Where the external heat transfer coefficient was obtained from the measured
average condenser temperature and heat input at each heat flux step. The condenser
99
temperature was measured at 16 locations on two radial axes at 0° and 90° from the
cooling air direction.
Figure 45: Condenser temperature distribution at the parallel and perpendicular radial axes –
analytical vs. experimental (Case 0).
The analytical model of the annular disc fin was solved using the commercial
software EES. The fin dimensions used are 1 in inner radius, 4.5 in outer radius, and 0.125
in fin thickness. Figure 45 shows the comparison between the analytical and experimental
condenser temperature distribution at the two radial axes. The comparison was made for
all the 15 total heat input values from 40 w to 600 w with 40 w steps. However, only three
power steps presented in the figure as the results showed a similar trend for all
100
configurations. For further validation, the analytical solution was compared to the
average condenser temperature (average of the 2 radial axes at each radius location); the
results are presented in Figure 46.
Figure 46: Radial condenser temperature distribution of empty heat pipe – analytical vs. average
experimental (Case 0).
The analytical model successfully predicted the average condenser temperature
within an error of ±2°, which is within temperature measurement uncertainty. However,
the temperatures at r = 3.75 in on both axes were out of the range. The recorded
temperature was almost constant at the initial condition, after a couple of tests, this was
noticed, and the diagnosis showed that the two connections through the DAQ were loose,
which gave unreliable readings for some cases.
101
The model was used to validate the thermal resistance model derived in Chapter
4.by obtaining the thermal resistance of the heat pipe under different filling ratios The
heat pipe thermal resistance in our study is strongly affected by external convection
resistance.
Figure 47 shows the experimental and theoretical thermal resistance values of the
heat pipe; the external conviction resistance is excluded from the Rhp to highlight the
effect of the external heat transfer coefficient. The experimental Rhp obtained using the
average measured temperatures on the evaporator and condenser. The one-phase heat
transfer regime, previously discussed in Figure 42, is apparent in the presented values.
At low heat input, the thermal resistance is relatively high, which is explained by the fact
that the only heat transfer path at this stage is the conduction between the evaporator and
condenser, as the two-phase heat transfer process would not be started at this stage. In
the heat pipe design under investigation, the evaporator is insulated from the condenser
using the Viton gasket in between, and that leaves the contact area is only the 1/8 in
thickness at the inner radius of the condenser plates. The developed model assumes
isothermal two-phase inside the heat pipe. Hence, The developed model does not predict
this region and highly underpredict the actual thermal resistance. The starting regime can
be reduced by decreasing the starting absolute pressure inside and eliminate the existence
102
of the non-condensable gases inside. [24]. The same curves are presented for the case of
90 ml filling of stainless-steel wick type in Figure 48.
Figure 47: Heat pipe effective and total thermal resistance for 120 ml charged heat pipe with
N=200 copper wick
Figure 48: Heat pipe effective and total thermal resistance for 90 ml charged heat pipe with
stainless steel wick
103
For further validation of the thermal resistance model, it was applied to the disc-
shaped heat pipe studied by North [80]. Using North’s published results, the total
thermal resistance was calculated based on the temperature difference between the
cooling water average and the evaporator center point temperature. Value of 5000W/m.
K was used for the external heat transfer coefficient. The use of water cooling reduced the
total thermal resistance, as illustrated in Figure 49. The dimensions of the North’s heat
pipe are described in detail in [81]. The model was used to calculate a single thermal
resistance value, due to the lack of the variating parameters. However, the predicted
value is within 5% of the obtained experimental value in the tow phase regime. Finally,
the results presented in this analysis solidified the level of confidence in the experimental
and analytical results obtained in this study. They provided the required validation of
the experimental design.
104
Figure 49: Thermal resistance model validation vs current experimental and North [81].
The design configuration of North’s heat pipe has the evaporator and part of the
condenser acting on the bottom plate; that radically reduced the conduction thermal
resistance. That explains the relatively small resistance in the one-phase regime
comparing to the design proposed in this study, as can be seen in Figure 49. This result
also agrees with the conduction performance curves obtained from the empty evacuated
cases for both designs presented in Figure 43. A schematic of the bottom plate in North’s
design is presented in Figure 50.
105
Figure 50: schematic of the fin analysis for North’s design - bottom condenser plate
Vapor Temperature
Following the discussion in Chapter 5, the assumption of isothermal vapor along
the heat pipe radial axis is will stated in the literature [29, 82]. The vapor temperature was
measured at three different locations along the radial axis to get a clear understanding of
the variation on the system, as illustrated in Figure 51.
Figure 51: Temperature locations in the vapor region
The vapor temperature measured at the center height of the vapor camper, at three
locations along the radial axis r = 1.25 in (location 1), 2.5 in (location 1), and 3.5 in (location
106
1). with r is the radius measured from the center of the disc heat pipe; that makes the first
point at 0.25 in from the evaporator surface. The measurement is taken for the Case 6D
with 600 w total heat input. Figure 52 shows the full recorded temperatures of the three
locations. The same data presented over 1 min average in Figure 53
Figure 52: Transient vapor pressure variation over the radial axis
107
Figure 53: One-minute average vapor temperature over the radial axis
The measured vapor temperatures show a good alignment between locations two
and three during the run period. The measured value at location 1 is higher by 3% of the
location 2 temperature in the startup and steady regions. The difference grows up to 5%
at the beginning of the cooling down period, enclosed in red dashed lines in Figure 53,
before it matches the other temperatures again at the close to flattening trend at the end
of the cooling-down period. This error is higher than the calibrated thermocouple’s
uncertainty of ±0.5. At location 1, the thermocouple is very close to the evaporator surface,
that area is expected to be the driest region in the heat pipe, as not much condensing
happens at the condenser plate close to the heat pipe center.
108
On the other hand, the thermocouples in locations 2 and 3 are in the active
condenser region, and it may be affected by the condensed water drops that could fall
from the top plate by the gravity forces. Figure 54 shows the top view case #1B, and the
dry region can be noticed around the evaporator centerpiece. This region is not
symmetrical around the center, and that’s due to the non-uniform cooling flow. It is also,
noticeable that there is more condensation accumulating on the working fluid charging
side of the heat pipe
Figure 54: Top view of the heat pipe with Acrylic top plate Case 1B
109
To investigate the effect of the existence of non-condensable gases inside the heat
pipe; the same test was repeated with different initial pressure at 1.5 kPa and 6.3 kPa by
vacuuming the heat pipe and then open the air vale carefully to just increase the pressure.
Worth mentioning that this process had to be repeated several times to achieve a
successful case, as it was hard to control the amount of air that enters the heat pipe once
the valve is slightly open. The results of the cases are compared to the theoretical
saturation pressure at the measured pressure and presented in
Figure 55: effect of non-condensable gases on the vapor region (run 1 @1.5 KPa, run 2
@6.3 KPa initial pressure)
110
Evaporator Temperature
The evaporator temperature is very critical in defining the annular disc-shaped
heat pipe under investigation. In the proposed design and application, it represents the
temperature of the steam condenser inner wall, which directly defines the condensing
temperature and pressure for the steam turbine. The evaporator temperature was
measured at 10 locations, and the exact locations are presented in Figure 30.
Thermocouples # 1,2, and 10 in Figure 30 are used to evaluate the thermal distribution
over the evaporator vertical axis. The three thermocouples are distributed at (Z/L = 1/2,
1/3, 2/3) at θ = 0, 10°, and 350°, respectively, where Z is measured from the top evaporator
surface and L = 1.25 in the total height of the cooper evaporator.
Figure 56: evaporator axial temperature distribution for case 4C
111
The average temperatures of the three locations for Case 6C heat pipe are plotted
against the total heat input in Figure 56. The error bars are generated from the average of
the three temperatures at each heat input. It is clear from the presented data the
isothermal behavior of the evaporator during the runs. However, the temperature error
reaches around 1.5°C at the high-power input of more than 600 w. this increase in the
temperature deviation indicates the increase in the axial conduction inside the
evaporator, which is a strong indication of reaching close to the heat pipe’s heat transfer
limit.
The 8 circumferential thermocouples (#1 and 4-10) are distributed at θ = 0, π/4, π/2,
3π/4, π, 5π/4, 3π/2, 7π/4, respectively. the circumferential temperature distribution for
the same case used in the axial temperature distribution evaluation is plotted in Figure
57
112
Figure 57: evaporator circumferential temperature distribution for case 4C
The evaporator has an explicit isothermal behavior at power input below 300 w.
However, the temperature deviation starts after that. The maximum recorded difference
is ±2.5 °C. this error is due to the non-uniform cooling on the condenser surface, as it was
discussed in Chapter 5. The thick sidewall and insulation cylinders cause recirculation on
the surface, which creates a discrepancy in the amount of cooling between the front and
back sides of the heat pipes. This explanation is justified by comparing the temperatures
of the front side (θ = 0, π/4, and 7π/4) to the backside (θ = 3 π /4, π, and 5π/4).
Condenser Temperature
To have the full characterization of a heat pipe thermal performance under
different heat loads and to accurately test its heat transfer limitation, it is recommended
113
to be tested with the capability of controlling the heat pipe working (vapor) temperature
[70]. Such a test could be achieved by employing active high capacity liquid cooling
methods, i.e., water or dry ice cooling. However, such working conditions does not imply
the real application of the proposed design under investigation. As explained, the
condenser temperature was measured on the top and bottom plates along two radial axes
at θ =0 and π/2. The temperature distribution over the top and bottom condenser for the
entire range of heat inputs for Case # 2D is presented in Figure 58 to Figure 61. As
expected, the temperature variation starts to take flat shape the farther we move away
from the evaporator. It is a justified assumption to consider the uniform temperature for
the condenser plates up to a total power of 500 w. however, at higher power values, the
temperature difference over the plate axis increase and reaches about 6°C. The maximum
variation in the temperature happens over the top plate axis at θ =0.
114
Figure 58: Radial temperature distribution of the bottom plate at θ = 0, (Case 2C)
Figure 59: Radial temperature distribution of the bottom plate at θ =π/2 (Case 2C)
115
Figure 60: Top plate radial temperature distribution at θ = 0 (Case 2C)
Figure 61: Top plate radial temperature distribution at θ = π/2 (Case 2C)
116
In general, it is noticed that the variation is higher on the axes that in line with the
flow (θ =0) comparing to the crossflow axes (θ = π/2). Figure 62 shows the average
temperatures of each axis and the total average over the top and bottom plates along with
the total average temperatures over the condenser considering all the surface area. The
Up-and-Down method described by Dixon [83] was adopted in this study to evaluate the
assumption of using the total average temperature for the condenser. the standard error
of the mean for the average temperature was calculated using equation (75)
𝑆𝐸𝑀 =𝑆𝐷
√𝑛 (75)
Where SD is the standard deviation of the samples, and n is the number of
samples. In our experiment, n = 24 as we have 24 thermocouples distributed over the
condenser plates. The calculated error for each input power is listed in Table 13. The
maximum calculated at the maximum power equal to 0.96
Figure 62: Surface average temperature of the top and bottom condenser plates vs. the
total heat input (Case 2C)
117
Table 13: Standard error of the condenser mean temperature
Q [w] Mean [°C] SD SEM
40 W 19.65954 0.467101 0.208894
80 W 20.60837 0.67418 0.301503
120 W 21.65522 0.814823 0.3644
160 W 22.46974 0.868578 0.38844
200 W 23.53013 0.919198 0.411078
240 W 24.57882 0.999848 0.447146
280 W 25.50234 1.128819 0.504823
320 W 26.41906 1.274685 0.570056
360 W 27.08954 1.345409 0.601685
400 W 28.02874 1.400084 0.626137
440 W 29.24575 1.554618 0.695246
480 W 30.10759 1.684263 0.753225
520 W 30.83722 1.77339 0.793084
560 W 31.71099 1.859096 0.831413
600 W 32.39506 1.982404 0.886558
640 W 33.35027 2.036062 0.910554
680 W 34.39917 2.085069 0.932471
700 W 34.9005 2.146479 0.959935
118
Thermal Resistance
The effective thermal resistance of the heat pipe is the characteristic parameter for
the heat pipe. Figure 63 presents the effective thermal resistance of the heat pipe
measured for different wick types at the same filling ratio.
Figure 63: variation of heat pipe effective thermal resistance with wick type
As expected, the stainless-steel wick has the highest thermal resistance, and the
wick with a smaller mesh number has the smallest effective thermal resistance. The
results show that the higher the total power, the lower resistance of the heat pipe. And
that’s due to the response time of the heat pipe to have the full steady operation, by
119
thoroughly having wetted evaporator without having an excessive amount of water in
the condenser, which adds more resistance to the system.
The effect of the filling ratio on the heat pipe effective thermal resistance is
illustrated in Figure 64. By comparing the three different filling ratios, Cases 3A (60 ml),
3B (90 ml), and 3C (120 ml); the small filling ratio has a higher thermal conductivity, and
this increase is more evident at high heat fluxes. However, excisive filling (120 ml) also
penalizes the thermal resistance at small heat fluxes. These results suggest that the
optimum filling amount for the heat pipe depends on its application and the amount of
heat flux applied to it.
Figure 64: ADHP effective thermal resistance at different filling ratios.
120
Figure 63 shows the variation of the ADHP thermal resistance with the change in
the heat input. The figure shows the results for Case 1B (90 ml) and 1C (120 ml), as
discussed before, the optimum filling ratio for the heat pipe depends on the heat input
operating range of the application.
Figure 65:ADHP thermal resistance variation with heat input. (case 1C and 1B)
121
CHAPTER 7: CONCLUSION AND FUTURE WORK
A novel air-cooled heat exchanger design has been proposed as an alternative
solution for the steam condenser in steam power plants. The proposed condenser design
implies the high thermal conductive heat pipe technology to reduce the total thermal
resistance between the internal steam and external cooling air. A novel annular disc-
shaped heat pipe has been designed to fit the application over steam tubes in the
condenser.
A detailed experimental setup has been designed to test the transient and steady
performance of the proposed heat pipe. The thermal performance and heat transfer
limitations of the heat pipe has been tested and compared to the theoretical model. The
heat pipe has been tested under different configurations of wick materials and working
fluid charging ratios. The experimental results showed a high impact of the external
cooling heat transfer coefficient on the heat pipe thermal resistance. And the filling ratio
has some influence on the heat pipe thermal performance, especially at low power input
conditions. The low filling ratios also have a longer startup response time. However,
excessive water in the heat pipe adds resistance to the condenser side, which also
penalizes the overall thermal performance. The experimental results suggest that the
optimal filling ratio for the designed heat pipe falls in the range of 7% to 10% of the total
heat pipe enclosed volume.
122
The heat transfer limits of the system have been addressed. A detailed derivation
of the boiling and capillary limits of the system have been presented. The experimental
results showed that heat transfer limits are profoundly affected by the existence of non-
condensable gases in the system. With 120 ml filing charge and 1.5 kPa initial vacuum
pressure, no dry-out indication noticed for up to 700 w of heat input. However, an
indication of dry-out was noticed above 400 w for the same configuration with 6.3 kPa
initial vacuum pressure.
A 3D numerical model has been derived in the study, Appendix A. The model
considers the heat transfer and hydrodynamics aspects of three regions of the heat pipe,
vapor, saturated wick, and solid regions. The model assumed only single-phase flow in
the vapor and wick regions. The phase change during condensation and evaporation is
modeled at the wick-vapor interface. The mass change in each phase was incorporated
as boundary conditions for each region. Heat source and sink were added to the wick
region energy equation to account for the latent heat of evaporation and condensation.
The inertial and viscous momentums in the liquid flow in the wick were considered by
adding momentum sources to the momentum equation of the liquid region. A
commercial software Star CCM+ was used to solve the model. The model showed a good
agreement with experimental results with the advantage of simplicity and time saving
compared to the modeling of the two-phase problem.
123
As a continuous work for this investigation, it is suggested to experimentally test
different sizes of the heat pipe, considering the ration of the evaporator to the condenser
surface ratio. The results of such an experiment would be needed to validate the model
further.
Identifying the heat transfer limitation of the system requires the experimental
setup to have the ability to control the condenser external heat transfer coefficient. Such
an experiment could be achieved by using temperature-controlled water flow as a cooling
medium, which would give the ability to increase the cooling capacity as increasing the
heat input to the evaporator and maintain the working vapor temperature constant. The
heat pipe limit could be identified by the heat input at where the evaporator temperature
starts to increase while the condenser temperature remains constant rapidly.
The developed CFD solution does not consider the effect of the condenser external
cooling. The experimental results showed the importance of the external heat transfer
coefficient and its effect on thermal performance. The model could be improved to
include condenser conditions. And to count for the non-symmetry conditions of the
system.
The CFD model could be used to conduct a parametric optimization study,
considering aspects of heat pipe dimensions, filling ratio, wick types, casing materials.
124
APPENDIX A: NUMERICAL ANALYSIS OF CONVENTIONAL
HEAT PIPE
125
NUMERICAL SETUP
Heat pipes employ phase change as the means to store and transfer heat energy.
The operational characteristics of the heat pipes make it a highly effective passive device
for transiting heat over small temperature difference [23]. Heat pipes have been utilized
in a wide range of application mainly in space satellites, its durability, passively
operation, and high conductivity, with the capability of design variation to fit custom
designs. Indeed, the most common use of heat pipes is in electronics cooling. Its high heat
transfer capability, and the ability to maintain constant evaporator temperature under a
relatively large range of heat flux, makes it a favorable choice as heat sinks for electronic
devices [24]. The growth in heat pipe usage in a variety of industrial applications raised
an essential need for more understanding of the heat pipe thermal characteristics and
performance limitations [59].
The numerical and analytical modeling of the heat pipes has progressed
significantly in the last few decades[59]. However, the physics phenomena involved in
the heat pipe operation make it challenging to have a comprehensive, detailed solution
for the system. Table 14 summarize the heat pipe operation and the interactions between
the heat pipe regions [23]. Many researchers have numerically modeled the heat pipe in
steady, transient, and start-up states. Garimella and Sobhan [84] have presented a review
of the recent advances in heat pipe modeling. The review is focused more on the non-
126
conventional heat pipes and its applications. Faghri [59] presented a comprehensive
review of the state-of-art works in heat pipes, covering experimental and modeling
advances in the field.
The thermal fluid phenomena in a heat pipe working process can be split into four
categories: (1) heat conduction in the solid outer container, (2) liquid flow in the wick
region, (3) vapor flow in the core region, and (4) the interaction between vapor and wick.
The first two categories have been extensively studied, many detailed analytical and
numerical models have been done. However, due to the complexity of the liquid flow in
the porous region, it is difficult to obtain an exact solution for the liquid flow in the wick
and the liquid-vapor interface region. That requires some practical information to be
extracted experimentally then used in the numerical modeling [26].
Figure 66: Schematic of the heat pipe and the coordinate system
127
Table 14: Heat pipe operation’s flow chart and its interaction between different regions.
Heat Source Heat Sink
Heat Conduction in heat pipe container
Variables Phenomena Governing Equations
Temperature
heat Input
Heat Output
Conjugate Effects
Boundary Conditions
Geometry Effect
Heat Conduction
Fourier Law
Liquid Flow and Heat Transfer in Wick Structure
Variables Phenomena Governing Equations
Temperature
Velocities
Pressure Drop
Gravity
Phase Change
Porous Media Flow
Interfacial Shear Stress
Liquid Entrainment
Boiling Limitation
Continuity Equation
Navier-Strokes Equations
Averaged Energy
Equation
Liquid-Vapor Interface
Variables Phenomena Governing Equations
Contact Angles
Interfacial Curvature
Interfacial Position
Interfacial Mass Fluxes
Capillary Pressure
Disjoining Pressure
Evaporation
Condensation
Interfacial Mass Balance
Interfacial Momentum
Balance
Interfacial Energy Balance
Clapeyron Equation
Vapor flow in heat pipe core
Variables Phenomena Governing Equations
Velocities
Pressure Drop
Density
Temperature
Concentration
Compressibility Effects
Continuum Flow
Mass Diffusion
Geometry Effects
Sonic Limitation
Continuity Equation
Momentum Equations
Energy Equation
Mass Diffusion Equation
Equation of State
128
Numerical Model
The finite volume method is used to discretize and solve the governing equations
of the fluid flow and heat transfer in the heat pipe. As shown in Figure 66, the cylindrical
heat pipe structurally consists of three sections: evaporator, adiabatic, and condenser
sections. However, for the mathematical modeling purpose, the heat pipe problem is
divided into core region (vapor), porous wick region (liquid), and the solid container
region. The governing equations for each region were derived along with its boundary
conditions and the coupling equations of the interfaces. The following common
assumptions are made in this simulation; these are usually made in analyzing heat pipes
[29, 74, 84-95]:
1. Vapor and liquid flows are steady, laminar, and subsonic.
2. Vapor flow simulated as an ideal gas.
3. Incompressible, non-Darcian transport flow in the wick region. (Wang and Ching
[91])
4. Axisymmetric around the heat pipe long axis.
5. Porous wick is always saturated with liquid-phase working fluid.
6. Single-phase (vapor) flow in the core region.
7. Single-phase (liquid) flow in the porous wick region.
129
8. Phase change processes (i.e., condensation and evaporation) occur at the wick-
vapor interface.
9. Vapor and liquid suction and injection rates are uniform over the evaporator and
condenser sections.
10. Radiative and gravitational effects are negligible.
Under the stated above assumptions, the 2D steady single-phase governing
equations (i.e., continuity, momentum, and energy equations.) for the vapor, wick, and
solid container regions are presented below.
Vapor Region
The continuity equation of the vapor flow in the core region is derived from the
conservation of mass of the incompressible single-phase flow in the region and
represented in equation (76). The vapor conservation of momentum in the axial and
radial directions are presented by Navier-Stokes equations (77) and (78). The energy
equation is derived by applying the conservation of energy to the vapor region, equation
(79).
𝜕𝑢
𝜕𝑥+
1
𝑟 𝜕(𝑟𝑣)
𝜕𝑟= 0 (76)
𝜌𝑣 (𝑣𝜕𝑢
𝜕𝑟+ 𝑢
𝜕𝑢
𝜕𝑥) = −
𝜕𝑝
𝜕𝑥+ 𝜇𝑣 (
𝜕2𝑢
𝜕𝑥2+
1
𝑟
𝜕𝑢
𝜕𝑟+
𝜕2𝑢
𝜕𝑟2) (77)
130
𝜌𝑣 (𝑣𝜕𝑣
𝜕𝑟+ 𝑢
𝜕𝑣
𝜕𝑥) = −
𝜕𝑝
𝜕𝑟+ 𝜇𝑣 (
𝜕2𝑣
𝜕𝑥2+
1
𝑟
𝜕𝑣
𝜕𝑟+
𝜕2𝑣
𝜕𝑟2−
𝑣
𝑟2) (78)
(𝑣𝜕𝑇
𝜕𝑟+ 𝑢
𝜕𝑇
𝜕𝑥) = 𝛼𝑣 (
𝜕2𝑇
𝜕𝑥2+
1
𝑟
𝜕𝑇
𝜕𝑟+
𝜕2𝑇
𝜕𝑟2) (79)
Porous-Wick Region
The governing equations of 2D steady-state of the liquid flow in the porous wick
region are presented by equations (80) to (83). The continuity equation of the liquid in the
porous region has the same form as the vapor region, equation (80). However, the
momentum equations in the porous region have extra momentum terms to present the
inertial and viscous resistance of the porous wick region, equations (81) and (82). As
mentioned above, the model considers the flow in the heat pipe regions as single phases.
So to account for the phase change process in the heat pipe, The latent heat of evaporation
and condensation processes is modeled by adding heat source and heat sink terms in the
wick condenser and evaporator regions, respectively, equation (83).
𝜕𝑢
𝜕𝑥+
1
𝑟 𝜕(𝑟𝑣)
𝜕𝑟= 0 (80)
𝜌𝑙 (𝑣𝜕𝑢
𝜕𝑟+ 𝑢
𝜕𝑢
𝜕𝑥) = −
𝜕𝑝
𝜕𝑥+ 𝜇𝑙 (
𝜕2𝑢
𝜕𝑥2+
1
𝑟
𝜕𝑢
𝜕𝑟+
𝜕2𝑢
𝜕𝑟2) −
𝜇𝑙휀
𝐾𝑢 −
𝐶𝐹
√𝐾|𝑢|𝑢 (81)
𝜌𝑙 (𝑣𝜕𝑣
𝜕𝑟+ 𝑢
𝜕𝑣
𝜕𝑥) = −
𝜕𝑝
𝜕𝑟+ 𝜇𝑙 (
𝜕2𝑣
𝜕𝑥2+
1
𝑟
𝜕𝑣
𝜕𝑟+
𝜕2𝑣
𝜕𝑟2−
𝑣
𝑟2) −
𝜇𝑙휀
𝐾𝑣 −
𝐶𝐹
√𝐾|𝑣|𝑣 (82)
131
(𝑣𝜕𝑇
𝜕𝑟+ 𝑢
𝜕𝑇
𝜕𝑥) = 𝛼𝑒𝑓𝑓 (
𝜕2𝑇
𝜕𝑥2+
1
𝑟
𝜕𝑇
𝜕𝑟+
𝜕2𝑇
𝜕𝑟2) + 𝑆𝑖 (83)
Solid Container
The heat transfer process in the heat pipe solid container is described by pure
conduction, as represented in equation (84)
(𝜕2𝑇𝑤𝑎
𝜕𝑥2+
1
𝑟
𝜕𝑇𝑤𝑎
𝜕𝑟+
𝜕2𝑇𝑤𝑎
𝜕𝑟2) = 0 (84)
CFD Model Validation
For the validation of the presented numerical model, the numerical simulations are
performed for a 1000 mm long cylindrical heat pipe with an outer diameter, wall
thickness, and wick thickness dimensions of 25.4 mm, 0.85 mm, and 0.356 mm,
respectively. The heat pipe axial sections have the dimensions of evaporator endcap 10
mm, evaporator 64 mm, an adiabatic section of 606 mm, and condenser section of 296
mm, and condenser endcap of 24 mm. The model results of the outside copper wall are
compared to the experimental results measured by Faghri [96] and the 3D numerical
results presented by Pooyoo [90]. Figure 67. Shows the comparison between the
measured and simulated results, a good agreement can be concluded from the
132
comparison. Ignoring the temperatures of the evaporator and condenser endcaps, the
maximum deviation between the simulated and measured data is 0.65°C.
Figure 67: CFD temperature distribution validation
133
APPENDIX B: WIND TUNNEL AIR-FLOW MEASUREMENTS
134
Table 15: Wind-Tunnel measurements
Unit Value
Height in 7.00
Width in 13.00
Average
temperature °C 22.80
kinematic viscosity m2/s 1.534 x 10-5
Total mass flow
rate m3/s 2.77
Average velocity m/s 41.4
Fan speed rpm 1428
Reynolds number
(Re) - 6.2 x105
Figure 68: Wind-Tunnel air-flow measurement grid
135
Figure 69: Wind-tunnel local traversed mass flow
Figure 70: Wind-tunnel local traversed mass flow
136
LIST OF REFERENCES
[1] Barker, B., "Running Dry at the Power Plant," EPRI Journal. 2007.
[2] Kenny, J. F., "Estimated Use of Water in the United States in 2005," Reston, Va : U.S.
Geological Survey, 2009.; 2009.
[3] Liner, B., Christopher Stacklin. "Driving Water and Wastewater Utilities to More
Sustainable Energy Management," POWER2013. Boston, Massachusetts, USA:
ASME; 2013.
[4] Macknick, J., Newmark, R., Heath, G., Hallett, K., "A Review of Operational Water
Consumption and Withdrawal Factors for Electricity Generating Technologies,"
Contract. 2011;303:275-3000.
[5] Meldrum, J. R., Averyt, K. B., Macknick, J. E., Newmark, R. L., Rogers, J., Madden, N.,
et al., "Sensitivities of Recent Electricity Generation Water Use Findings to Data
Updates and Variability," ASME Power Conference. Boston, Massachusetts,
USA2013.
[6] Meldrum, J. R., Macknick, J. E., Heath, G. A., Nettles-Anderson, S. L., "Life Cycle Water
Use for Electricity Generation: Implications of the Distribution of Collected
Estimates," POWER2013. Boston, Massachusetts, USA: ASME; 2013.
137
[7] Micheletti, W. C., Burns, J. M., "Emerging Issues and Needs in Power Plant Cooling
Systems," Proceedings of the Workshop on Electric Utilities and Water: Emerging
issues and R&D needs: Pittsburgh, Pennsylvania2002.
[8] Tidwell, V., Gasper, J., Goldstein, R., Macknick, J., Sehlke, G., Webber, M., et al.,
"Integrated Energy-Water Planning in the Western and Texas Interconnections,"
POWER2013. Boston, Massachusetts, USA: ASME; 2013.
[9] Tidwell, V. C., Kobos, P. H., Malczynski, L. A., Klise, G., Castillo, C. R., "Exploring the
Water-Thermoelectric Power Nexus," Journal of Water Resources Planning &
Management. 2012;138:491-501.
[10] Torcellini, P. A., Long, N., Judkoff, R. D., "Consumptive Water Use for U.S. Power
Production," ASHRAE Transactions. 2004;110:96-100.
[11] Yan, Y. E., Demissie, Y. K., Wigmosta, M. S., Tidwell, V. C., King, C. W., Cook, M. A.,
"Potential Drought Impacts on Electricity Generation in Texas," POWER2013.
Boston, Massachusetts, USA: ASME; 2013.
[12] Grinnan, M. H., "Experience Replacing Feedwater Heaters and Condensers Due to
Repowering and Uprating a Power Plant," POWER2013. Boston, Massachusetts, USA:
ASME; 2013.
[13] Card, R. W., "Economic Design of Hibrid Wet-Dry Cooling System," ASME 2013
Power Conference. Boston, Massachusetts, USA: ASME; 2013.
138
[14] Ismayilov, F., Wang, Y., Peles, Y., "Challenges in Micro-Channel Heat Transfer
Experiments: Insight on Conjugate Heat Transfer Effects," 2017 16th IEEE
Intersociety Conference on Thermal and Thermomechanical Phenomena in
Electronic Systems (ITherm): IEEE; 2017. p. 263-275.
[15] Wilber, K., "Air-Cooled Condenser Design, Specification, and Operation Guidelines,"
EPRI, Palo Alto, CA: 2005.
[16] Sonntag, R. E., Borgnakke, C., Van Wylen, G. J., Van Wyk, S., "Fundamentals of
Thermodynamics," Wiley New York; 1998.
[17] Shi, J., Bushart, S., "Development of Potential Game Changing Water Conserving
Cooling Technologies," Cooling Tower Technology Conference. EPRI, Palo Alto,
CA; 2012.
[18] Conradie, A. E., Kröger, D. G., "Performance Evaluation of Dry-Cooling Systems for
Power Plant Applications," Applied Thermal Engineering. 1996;16:219-232.
[19] Bustamante, J. G., Rattner, A. S., Garimella, S., "Achieving near-Water-Cooled Power
Plant Performance with Air-Cooled Condensers," Applied Thermal Engineering.
2016;105:362-371.
[20] Owen, M., Kröger, D. G., "A Numerical Investigation of Vapor Flow in Large Air-
Cooled Condensers," Applied Thermal Engineering. 2017;127:157-164.
139
[21] Team, C., "Mathematical Modeling, Numerical Simulation and Statistical
Optimization of Heat Pipe Design." 2006.
[22] Zohuri, B., "Heat Pipe Design and Technology: Modern Applications for Practical
Thermal Management," Springer; 2016.
[23] Faghri, A., "Heat Pipe Science and Technology," Global Digital Press; 1995.
[24] Peterson, G. P., "An Introduction to Heat Pipes : Modeling, Testing, and Applications
/ G.P. Peterson," New York : Wiley, c1994.; 1994.
[25] Silverstein, C. C., "Design and Technology of Heat Pipes for Cooling and Heat
Exchange / Calvin C. Silverstein," Washington : Hemisphere Pub. Corp., c1992.;
1992.
[26] Zohuri, B., "Heat Pipe Design and Technology : A Practical Approach / Bahman
Zohuri," Boca Raton, FL : CRC Press, 2011.; 2011.
[27] Elsayed, M. L., Mesalhy, O., "Studying the Performance of Solid/Perforated Pin-Fin
Heat Sinks Using Entropy Generation Minimization," Heat and Mass transfer.
2015;51:691-702.
[28] Yang, X., Yan, Y. Y., Mullen, D., "Recent Developments of Lightweight, High
Performance Heat Pipes," Applied Thermal Engineering. 2012;33–34:1-14.
[29] Vafai, K., Zhu, N., Wang, W., "Analysis of Asymmetric Disk-Shaped and Flat-Plate
Heat Pipes," Journal of Heat Transfer. 1995;117:209-218.
140
[30] O'Donovan, A., Grimes, R., "A Theoretical and Experimental Investigation into the
Thermodynamic Performance of a 50 mw Power Plant with a Novel Modular Air-
Cooled Condenser," Applied Thermal Engineering. 2014;71:119-129.
[31] Saleh, A., Kapat, J., "Comprehensive Reduction of Thermal Resistance in Air Cooled
Condensers," ASME 2015 Power Conference2015. p. V001T004A002.
[32] Mills, A. F., "Basic Heat and Mass Transfer," Prentice hall; 1999.
[33] Critoph, R. E., Holland, M. K., Turner, L., "Contact Resistance in Air-Cooled Plate
Fin-Tube Air-Conditioning Condensers," International Journal of Refrigeration.
1996;19:400-406.
[34] Kim, C. N., Jeong, J., Youn, B., "Evaluation of Thermal Contact Conductance Using a
New Experimental-Numerical Method in Fin-Tube Heat Exchangers,"
International Journal of Refrigeration. 2003;26:900-908.
[35] ElSherbini, A. I., Jacobi, A. M., Hrnjak, P. S., "Experimental Investigation of Thermal
Contact Resistance in Plain-Fin-and-Tube Evaporators with Collarless Fins,"
International Journal of Refrigeration. 2003;26:527-536.
[36] Eckartsberg, A., Kapat, J., Effect of Flow Turbulence on Evaporation Rate of a Single
Droplet. 38th Aerospace Sciences Meeting and Exhibit: American Institute of
Aeronautics and Astronautics; 2000.
141
[37] Stewart, S. W., "Enhanced Finned-Tube Condenser Design and Optimization,"
[Ph.D.], Georgia Institute of Technology; Ann Arbor 2003.
[38] O’Donovan, A., Grimes, R., Moore, J., "The Influence of the Steam-Side
Characteristics of a Modular Air- Cooled Condenser on Csp Plant Performance,"
Energy Procedia. 2014;49:1450-1459.
[39] Hu, H. G., "Simulation of Two-Phase Flow and Heat Transfer in Condensers,"
[Ph.D.], The University of Western Ontario (Canada); Ann Arbor 2008.
[40] O’Donovan, A., Grimes, R., Walsh, E., Moore, J., Reams, N., "Steam-Side
Characterisation of a Modular Air-Cooled Condenser," ASME 2012 International
Mechanical Engineering Congress and Exposition: American Society of
Mechanical Engineers; 2012. p. 1719-1730.
[41] Kayansayan, N., "Heat Transfer Characterization of Plate Fin-Tube Heat
Exchangers," International Journal of Refrigeration. 1994;17:49-57.
[42] Keyes, R. W., "Heat Transfer in Forced Convection through Fins," Electron Devices,
IEEE Transactions on. 1984;31:1218-1221.
[43] Bilirgen, H., Dunbar, S., Levy, E. K., "Numerical Modeling of Finned Heat
Exchangers," Applied Thermal Engineering. 2013;61:278-288.
142
[44] Li, L., Du, X., Yang, L., Xu, Y., Yang, Y., "Numerical Simulation on Flow and Heat
Transfer of Fin Structure in Air-Cooled Heat Exchanger," Applied Thermal
Engineering. 2013;59:77-86.
[45] Elsayed, M. L., Mesalhy, O., Kizito, J. P., Leland, Q. H., Chow, L. C., "Performance of
a Guided Plate Heat Sink at High Altitude," International Journal of Heat and Mass
Transfer. 2020;147:118926.
[46] Elsayed, M. L., Mesalhy, O., Mohammed, R. H., Kizito, J. P., Leland, Q. H., Chow, L.
C., "Enhancement of a Heat Sink Performance Using a Partial Shield and/or a
Guide Plate," International Journal of Heat and Mass Transfer. 2019;134:948-958.
[47] Elsayed, M. L., Mesalhy, O., "Effect of a Slotted Shield on Thermal and Hydraulic
Performance of a Heat Sink," Journal of Electronic Packaging. 2015;137.
[48] Shaalan, M. R., Saleh, M. A., Mesalhy, O., Elsayed, M. L., "Thermo/Fluid Performance
of a Shielded Heat Sink," International Journal of Thermal Sciences. 2012;60:171-
181.
[49] Sparrow, E., "Analysis of Laminar Forced-Convection Heat Transfer in Entrance
Region of Flat Rectangular Ducts," National Advisory Committee for Aeronautics;
1955.
[50] Teertstra, P., Yovanovich, M. M., Culham, J. R., Lemczyk, T., "Analytical Forced
Convection Modeling of Plate Fin Heat Sinks," Semiconductor Thermal
143
Measurement and Management Symposium, 1999 Fifteenth Annual IEEE1999. p.
34-41.
[51] Shah, R. K., Sekulic, D. P., "Fundamentals of Heat Exchanger Design," John Wiley &
Sons; 2003.
[52] Churchill, S. W., Usagi, R., "A General Expression for the Correlation of Rates of
Transfer and Other Phenomena," AIChE Journal. 1972;18:1121.
[53] Moore, J., Grimes, R., Walsh, E., O'Donovan, A., "Modelling the Thermodynamic
Performance of a Concentrated Solar Power Plant with a Novel Modular Air-
Cooled Condenser," Energy. 2014;69:378-391.
[54] Kröger, D. G., "Air-Cooled Heat Exchangers and Cooling Towers / by Detlev G.
Kröger," Tulsa, Okl. : Penwell Corp., c2004.; 2004.
[55] Incropera, F. P., "Fundamentals of Heat and Mass Transfer," John Wiley & Sons; 2011.
[56] Marcus, B. D., "Theory and Design of Variable Conductance Heat Pipes," National
Aeronautics and Space Administration; 1972.
[57] Ababneh, M. T., Gerner, F. M., Chamarthy, P., Bock, P. d., Chauhan, S., Deng, T.,
"Thermal-Fluid Modeling for High Thermal Conductivity Heat Pipe Thermal
Ground Planes," Journal of Thermophysics and Heat Transfer. 2014;28:270-278.
[58] Faghri, A., "Review and Advances in Heat Pipe Science and Technology," Journal of
Heat Transfer. 2012;134.
144
[59] Faghri, A., "Heat Pipes: Review, Opportunities and Challenges," Frontiers in Heat
Pipes (FHP). 2014;5.
[60] Asselman, G., Green, D., "Heat Pipes," Philips Tech Rev. 1973;16:169-186.
[61] Rayleigh, L., "Lvi. On the Influence of Obstacles Arranged in Rectangular Order
Upon the Properties of a Medium," The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science. 1892;34:481-502.
[62] Chi, S., "Heat Pipe Theory and Practice,"1976.
[63] Mantle, W. J., Chang, W. S., "Effective Thermal Conductivity of Sintered Metal
Fibers," Journal of Thermophysics and Heat Transfer. 1991;5:545-549.
[64] Şen, T., "Two Dimensional Numerical Modeling of a Flat Heat Pipe," İzmir Institute
of Technology; 2014.
[65] Shabgard, H., Faghri, A., "Performance Characteristics of Cylindrical Heat Pipes with
Multiple Heat Sources," Applied Thermal Engineering. 2011;31:3410-3419.
[66] Mills, A. F., "Heat Transfer," Irwin; 1992.
[67] Ababneh, M., "Novel Charging Station and Computational Modeling for High
Thermal Conductivity Heat Pipe Thermal Ground Planes," University of
Cincinnati; 2012.
[68] Dunn, P. D., Reay, D., "Heat Pipes," Elsevier; 2012.
145
[69] Lefèvre, F., Conrardy, J.-B., Raynaud, M., Bonjour, J., "Experimental Investigations of
Flat Plate Heat Pipes with Screen Meshes or Grooves Covered with Screen Meshes
as Capillary Structure," Applied Thermal Engineering. 2012;37:95-102.
[70] Tsai, T.-E., Wu, G.-W., Chang, C.-C., Shih, W.-P., Chen, S.-L., "Dynamic Test Method
for Determining the Thermal Performances of Heat Pipes," International Journal
of Heat and Mass Transfer. 2010;53:4567-4578.
[71] Jankowski, T. A., "Numerical and Experimental Investigations of a Rotating Heat
Pipe," Los Alamos National Lab.(LANL), Los Alamos, NM (United States): 2007.
[72] Huang, L., El‐Genk, M. S., "Experimental Investigation of Transient Operation of a
Water Heat Pipe," AIP Conference Proceedings: American Institute of Physics;
1993. p. 365-374.
[73] Tien, C., "Fluid Mechanics of Heat Pipes," Annual review of fluid mechanics.
1975;7:167-185.
[74] Zhu, N., Vafai, K., "The Effects of Liquid-Vapor Coupling and Non-Darcian
Transport on Asymmetrical Disk-Shaped Heat Pipes," International Journal of
Heat and Mass Transfer. 1996;39:2095-2113.
[75] Mohammed, R. H., Mesalhy, O., Elsayed, M. L., Chow, L. C., "Assessment of
Numerical Models in the Evaluation of Adsorption Cooling System Performance,"
International Journal of Refrigeration. 2019;99:166-175.
146
[76] Mesalhy, O., Elsayed, M. L., Corona Jr, J. J., Kwarteng, A. A., Kizito, J. P., Leland, Q.
H., et al., "Study of a High-Reliability Dual-Fan System for Cooling Aerospace
Electromechanical Actuators," Thermal Science and Engineering Progress.
2020:100540.
[77] Figliola, R. S., Beasley, D. E., "Theory and Design for Mechanical Measurements,"
IOP Publishing; 2001.
[78] Moffat, R. J., "Describing the Uncertainties in Experimental Results," Experimental
Thermal and Fluid Science. 1988;1:3-17.
[79] Otto, M., "Improving Turbine Performance: A Contribution to the Understanding of
Heat Transfer and Vortical Structures in Staggered Pin Fin Arrays." 2019.
[80] North, M., Avedisian, C., "Heat Pipes for Cooling High Flux/High Power
Semiconductor Chips," Journal of Electronic Packaging. 1993;115:112-117.
[81] North, M. T., "Heat Pipes for Cooling High-Flux/High-Power Semiconductor
Devices," [Ph.D.], Cornell University; Ann Arbor 1993.
[82] Tournier, J.-M., El-Genk, M., "A Heat Pipe Transient Analysis Model," International
Journal of Heat and Mass Transfer. 1994;37:753-762.
[83] Dixon, W. J., "Efficient Analysis of Experimental Observations," Annual review of
pharmacology and toxicology. 1980;20:441-462.
147
[84] Garimella, S. V., Sobhan, C., "Recent Advances in the Modeling and Applications of
Nonconventional Heat Pipes," Advances in Heat Transfer. 2001;35:249-308.
[85] Aghvami, M., Faghri, A., "Analysis of Flat Heat Pipes with Various Heating and
Cooling Configurations," Applied Thermal Engineering. 2011;31:2645-2655.
[86] Brahim, T., Jemni, A., "Heat Pipe Simulation under Critical Conditions," Frontiers in
Heat Pipes. 2013;3.
[87] Faghri, A., "Recent Advances in the Numerical Analysis of Heat Pipes,"
Computational Mechanics. 1994;14:480-491.
[88] Kaya, T., Goldak, J., "Three-Dimensional Numerical Analysis of Heat and Mass
Transfer in Heat Pipes," Heat and Mass transfer. 2007:775.
[89] Mahdavi, M., Qiu, S., "Mathematical Modeling and Analysis of Steady State
Performance of a Heat Pipe Network," Applied Thermal Engineering. 2015;91:556-
573.
[90] Pooyoo, N., Kumar, S., Charoensuk, J., Suksangpanomrung, A., "Numerical
Simulation of Cylindrical Heat Pipe Considering Non-Darcian Transport for
Liquid Flow inside Wick and Mass Flow Rate at Liquid–Vapor Interface,"
International Journal of Heat and Mass Transfer. 2014;70:965-978.
[91] Wang, C., Cheng, P., Multiphase Flow and Heat Transfer in Porous Media. Advances
in Heat Transfer: Elsevier; 1997. p. 93-196.
148
[92] Wang, Y., Vafai, K., "An Experimental Investigation of the Thermal Performance of
an Asymmetrical Flat Plate Heat Pipe," International Journal of Heat and Mass
Transfer. 2000;43:2657-2668.
[93] Yue, C., Zhang, Q., Zhai, Z., Ling, L., "Cfd Simulation on the Heat Transfer and Flow
Characteristics of a Microchannel Separate Heat Pipe under Different Filling
Ratios," Applied Thermal Engineering. 2018;139:25-34.
[94] Zhu, N., Vafai, K., "Analytical Modeling of the Startup Characteristics of
Asymmetrical Flat-Plate and Diskshaped Heat Pipes," International Journal of
Heat and Mass Transfer. 1998;41:2619-2637.
[95] Zhu, N., Vafai, K., "Analysis of Cylindrical Heat Pipes Incorporating the Effects of
Liquid–Vapor Coupling and Non-Darcian Transport—a Closed Form Solution,"
International Journal of Heat and Mass Transfer. 1999;42:3405-3418.
[96] Faghri, A., "Analysis of Frozen Startup of High-Temperature Heat Pipes and Three-
Dimensional Modeling of Block-Heated Heat Pipes," Report, Dept of Mechanical
and Materials Engineering, Wright State Univ. , Dayton, OH: 1991.